Finding Numerical Derivatives. While the TI-83+/84+ calculators are not symbolic manipulators (they cannot tell you the algebraic expression for a derivative) Find the numerical value of the derivative for the function stated below at the indicated location (express answer to three decimal places). Graphs of Derivatives – Discovery: This three-page worksheet will guide your students to graph the derivative of a function and make observations about the following concepts: * The slope of a tangent line to a curve can be identified at various points and used to create the graph of the derivativ.

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## Calc Finding the Derivatives of Trig Functions Notes | Quizlet

How to use the chain rule to find the derivatives of a trig. function? Put the inside piece of the function into the derivative of the previous outside function, then multiply by the derivative of the next outside function. Repeat these steps for a multi step chain rule problem.It is used to compute derivatives of simple expressions. Parameters: expr: represents an expression or a formula with no LHS name: represents character vector to which derivatives will be computed.Tool to find the equation of a function from its points, its coordinates x, y=f(x) according to some interpolation methods and equation finder algorithms. Answers to Questions (FAQ).How to graph Reciprocal Functions, characteristics of graphs of reciprocal functions, use transformations to graph a reciprocal function, how The graph of y = gets closer to the x-axis as the value of x increases, but it never meets the x-axis. This is called the horizontal asymptote of the graph.

Finding Numerical Derivatives. While the TI-83+/84+ calculators are not symbolic manipulators (they cannot tell you the algebraic expression for a derivative) Find the numerical value of the derivative for the function stated below at the indicated location (express answer to three decimal places).Graphs of Derivatives – Discovery: This three-page worksheet will guide your students to graph the derivative of a function and make observations about the following concepts: * The slope of a tangent line to a curve can be identified at various points and used to create the graph of the derivativ.This free slope calculator solves for multiple parameters involving slope and the equation of a line.Derivatives of logarithmic functions are simpler than they would seem to be, even though the functions themselves come from an important limit in Calculus. and I don’t know how to deal with it. Example (Click to try) 2 x 2 − 5 x − 3. Derivative Calculator.

## Calculus/Differentiation/Differentiation Defined – Wikibooks, open books

Calculus/Differentiation/Differentiation Defined. Language. Watch. Edit. < Calculus | Differentiation. x2-4x+3÷x2-5x+6. Differentiation is a process of finding a function that outputs the rate of change of one variable with respect to another variable.How to Find the Y-Intercept of a Line of a Graph in Excel on a Mac. on the screen. A derivative basically finds the slope of a function..To find the derivative at a point we can draw the tangent line to the graph of a cubic function at that point The integral concept is associate to the concept of area. We began considering the area limited by the graph of a function and the x-axis between two vertical lines.The purpose of finding a derivative is to find the instantaneous rate of change. To find the inverse of L, which is a lower triangular matrix, you can find the answer in this link.www.mcs.csueastbay.edu/~malek/TeX/Triangle.

How can I find the function if I know its derivative? This guide is meant to provide one with the tools one will need to calculate derivatives of basic functions.To find the derivative of a function y = f(x) we use the slope formula: Slope = Change in Y Change in X = ΔyΔx. And (from the diagram) we see that: Now follow these steps: Fill in this slope formula: ΔyΔx = f(x+Δx) − f(x)Δx.IEstimating the derivative of a function from a graph is an important skill for math and science students, and it works well provided you can draw an accurate tangent line to the point on the Take the slope of this line to find the value of the derivative at your chosen point on the graph.

## Derivative Graphs – Polynomials | Things to do

In this interactive graph, you can explore the concept of derivatives (calculus) of polynomials. In the following interactive you can explore how the slope of a curve changes as the variable `x` changes. Things to do. In this applet, there are pre-defined examples in the pull-down menu at the top.In calculus, you need to graph the derivative of a function in order to find its critical points, which you can do on your TI-84 Plus calculator. If necessary, repeatedly press the up- and down-arrow keys until the derivative function appears in the border at the top of the screen.Stack Exchange network consists of 178 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. $ I’m trying to get the derivative of this functional energy, to verify that it’s nonincreasing.How To Find The Derivative Of An Inverse Function. If f(x) is a continuous one-to-one function defined on an interval, then its inverse is also continuous.

Background. -Derivatives. The derivative of a function is a representation of how it changes. Google Sheets supports over 300 functions. A comprehensive list can be found here: https Derivatives are fundamental to calculus and physics.Use our simple online Derivative Calculator to find derivatives with step-by-step explanation. You can calculate partial, second, third, fourth derivatives as well as antiderivatives with ease and for free. Building graphs and using Quotient, Chain or Product rules are available.Taking derivative: The graph of the curve is represented below and the slope is represented in red line Therefore stableswap combines both the methods in order to find the middle way where pools would not run out of the assets nor do the traders have to face expensive trading.These derivatives correspond to each of the independent variables and can be interpreted as instantaneous rates of Now we consider the possibility of a tangent line parallel to neither axis. Directional Derivatives. Figure 4.39 Finding the directional derivative at a point on the graph of.

## Online Derivative Calculator | Systems of equations

This is a calculator which computes derivative, minimum and maximum of a function with respect to a variable x. This calculator evaluates derivatives using analytical differentiation. It will also find local minimum and maximum, of the given function.Several examples with detailed solution on how to find the derivatives of functions are presented along with detailed solutions. Several Examples with detailed solutions are presented. More exercises with answers are at the end of this page.To sketch the graph of a function, we need to perform the following After you’ve found “important” points calculate corresponding values of function at these points. Clearly they are all points of inflections because second derivative changes sign at these points.B

The graph of a derivative of a function is related to the graph of . Found inside – Page 3875.5 Exercises In Exercises 1-34 Explain how the sign of the second derivative affects the shape of a function’s graph. Use the vertical line test to determine whether or not a graph represents a function.Derivative rules and laws. Derivatives of functions table. The derivative of a function is the ratio of the difference of function value f(x) at points x+Δx and x with Δx, when Δx is infinitesimally small.How is the derivative related to the integral of a function? How do you find an integral? After the Integral Symbol we put the function we want to find the integral of (called the Integrand). And then finish with dx to mean the slices go in the x direction (and approach zero in width).To compute the second derivative, just take the differences of the first derivative values, divide by the differences of the midpoint volumes and plot this at the point between the two midpoint volumes. The following spreadsheet example should make this clear.

## EXCEL How To Make A Titrimetric Curve, 1st Derivative Plot And 2nd Derivative Plot 442021

This demonstration is on how to create a titrimetric curve, a first derivative curve and a second derivative curve. Here is a set of data whereone is plotting a weak monoprotic acid vs a standardized sodium hydroxide solution. The molarity of this acid is 0. 1043 molar and a 20-milliliter aliquot is titrated witha solution of sodium hydroxide whose molarity is 0. 1025 molar. Here is a set of data from0 to 30 milliliters with their corresponding phs. For column c, one takes in cell c3 andenters equals a3 minus a2 and press enter. For column d, one takes in cell d3 and entersequals b3 minus b2 and press enter. To complete the remainder of columns c and d, highlightcells c3 and d3 and in the bottom right-hand corner, use the small black plus in cell d3,click and drag down the rest of the cells to be computed. for column e, to calculatethe average volume, you type in cell e3 average of a2 and a3 in parentheses and press enter. For column f, type in cell f3 equal d3 and divide by c3 and press enter.

highlight columnse3 and d3 and move to the lower right hand corner in cell f3, click and drag the blackplus down. for column g, the calculations are similar to e3 so i will use the same formulaonly averaging cells a3 and a4 in cell g4 and press enter. for column h in performingthe values in d squared ph over dv squared, enter in cell h4 equals f4 divided by c4 andpress enter. Now highlight g4 and h4 and in the bottom right hand corner of cell h4, clickand drag the black plus downward. now you make a titrimetric graph between columns aand b, a first derivative plot between columns e and f and a second derivative plot betweencolumns g and h. To make the titrimetic curve, highlight thevalues of a and b, click on the insert tab and select under charts scatter with smoothlines and markers. Next, under the design tab, click under chart layouts quick layoutand select layout 1. Under chart styles, select a layout where the x and y gridlines are visible. if needed, you can change the font and size by clicking the home tab. label all axes andprovide an appropriate title.

the title of this graph will be titrimetric curve of havs. Naoh. The y-axis is labeled ph and the x-axis is labeled volume of naoh in millilitersin parentheses. perform a similar approach for the other graphs and here are their graphsshown here. for the titrimetric curve, type in all half and equivalence points by specifyingthe volumes and pka at each half-equivalence point. For the first derivative plot, youwant to label all apex where all maximums are observed and label its volume. For thesecond derivative plot, label all volumes where the curve passes through the x-axisgoing from a y maximum to a y-minimum in value and label all volumes.

. This demonstration is on how to create a titrimetric curve, a first derivative curve and a second derivative curve. Here is a set of data whereone is plotting a weak monoprotic acid vs a standardized sodium hydroxide solution. The molarity of this acid is 0. 1043 molar and a 20-milliliter aliquot is titrated witha solution of sodium hydroxide whose molarity is 0. 1025 molar. Here is a set of data from0 to 30 milliliters with their corresponding phs. For column c, one takes in cell c3 andenters equals a3 minus a2 and press enter. For column d, one takes in cell d3 and entersequals b3 minus b2 and press enter. To complete the remainder of columns c and d, highlightcells c3 and d3 and in the bottom right-hand corner, use the small black plus in cell d3,click and drag down the rest of the cells to be computed. for column e, to calculatethe average volume, you type in cell e3 average of a2 and a3 in parentheses and press enter. For column f, type in cell f3 equal d3 and divide by c3 and press enter.

highlight columnse3 and d3 and move to the lower right hand corner in cell f3, click and drag the blackplus down. for column g, the calculations are similar to e3 so i will use the same formulaonly averaging cells a3 and a4 in cell g4 and press enter. for column h in performingthe values in d squared ph over dv squared, enter in cell h4 equals f4 divided by c4 andpress enter. Now highlight g4 and h4 and in the bottom right hand corner of cell h4, clickand drag the black plus downward. now you make a titrimetric graph between columns aand b, a first derivative plot between columns e and f and a second derivative plot betweencolumns g and h. To make the titrimetic curve, highlight thevalues of a and b, click on the insert tab and select under charts scatter with smoothlines and markers. Next, under the design tab, click under chart layouts quick layoutand select layout 1. Under chart styles, select a layout where the x and y gridlines are visible. if needed, you can change the font and size by clicking the home tab. label all axes andprovide an appropriate title.

the title of this graph will be titrimetric curve of havs. Naoh. The y-axis is labeled ph and the x-axis is labeled volume of naoh in millilitersin parentheses. perform a similar approach for the other graphs and here are their graphsshown here. for the titrimetric curve, type in all half and equivalence points by specifyingthe volumes and pka at each half-equivalence point. For the first derivative plot, youwant to label all apex where all maximums are observed and label its volume. For thesecond derivative plot, label all volumes where the curve passes through the x-axisgoing from a y maximum to a y-minimum in value and label all volumes.

.

## How To Find The Equation Of A Tangent Line With Derivatives (NancyPi)

Hi guys! I'm nancy.. And i'm going to show you how to find the equation of a tangent line using derivatives. What is a tangent line? Well, it just means if you have a graph… With some curve on it the tangent line is the line that touches your graph at one point exactly. It's a linear line that is tangent to your curve and sort of grazes it and touches at one point. So tangent line problems can come in many forms but thankfully you can use the same four steps in all of them so let me show you some examples.

ok. Say you have a problem that says to find the tangent line to y equals x cubed plus 2x at a point at the point: 1, 3. How do you start this problem? Well it's always gonna be the same four steps in these tangent line problems. The first step is to take the derivative of the equation that you're given. This y equals is also f of x.

it's also a function. You want to take the derivative of the equation you're given, so f prime of x is your first step. F prime of x and then you take the derivative of this expression. The derivative of x cubed… Remember for the power rule, you bring the power down and then decrease that power by 1.

so for the first term, we have 3x squared for the second term, the derivative of 2x is just 2. The x goes away. You have 3x squared plus 2. So that's your derivative, your f prime of x. The second step is always to take the x coordinate that you're given in the point…

and plug it in to f prime of x. You can call that f prime of 1. Since you're x coordinate is 1. So you take that number, and you're going to plug it in to what you just found, the derivative. And this is just notation for that, for plugging in 1.

anyway, when you plug in 1 for x, you have 3 times 1 squared plus 2… Which simplifies to 3 times 1 plus 2 which is just 5, so f prime of 1… Equals 5. This is going to be your slope number that you'll use. The third step is…

to find the y coordinate of your point, if you're not already given it so sometimes you're not given the y. In this problem, you're given the whole point, which could happen. So you have y equals 3, so you don't need to find y in this case. You can move on to the fourth step, the last step, is to plug in the numbers that you've found into the point-slope formula for a line. The point-slope formula…

in general looks like y minus y1 equals m times x minus x1. M is just your slope and that's going to be the f prime value that you found when you plugged in x so your m in this problem is 5. Which you found right here. X1 and y1 are the coordinates of the point that you found or that you were given. So this here you can also call x1, y1.

you have both of those numbers. Now you want an equation for the tangent line. That means that in your equation you need to leave an x variable and y variable in it. So it's ok to leave those there. In fact you want it.

you can go ahead and you can write your equation by plugging in values for m, for x1, and for y1. So your x1 is the coordinate 1. Your y1 is the y coordinate of the point. The original point, which is 3. So if you rewrite this, your equation is y – 3 equals the slope 5, times (x – 1).

notice you're leaving x and y in the equation because you want it to be an equation for y in terms of x. So this… This is correct. This is a correct answer. Sometimes it's considered simpler to move it around and change it to be in slope intercept form or y = mx + b form.

so you can do that with some simple algebra if you were to multiply, distribute out, the… 5x – 5, and rearrange the numbers you could get into y equals form. If you were to multiply, distribute out, the… 5x – 5, and rearrange the numbers you could get into y equals form. Y = 5x – 5 + 3… Which is just minus 2.

so this.. Y = 5x – 2 is your equation of the tangent line to this f of x at that point. Alright, say you have f(x) = x / (x – 1) and you have to find the equation of the tangent line at x = 2. This looks a lot more complicated but the truth is it's the same four steps no matter how complicated your f(x), your equation gets. But the truth is it's the same four steps no matter how complicated your f(x), your equation gets. This could be ridiculous..

but as long as you know how to take f prime of x, the derivative, you can still do all the same steps and get the equation of the linear tangent line. So the first step is still, find f of x. Find the derivative of your given equation. You want f prime of x.. Now since your equation is x / (x – 1) this is a fraction.

this is a quotient. So you'll have to use the quotient rule to take the derivative. Which is a little more complicated than you might like.. But the steps are pretty straightforward. Think back to doing the quotient rule..

it's the bottom of the function times the derivative of the top.. Derivative of x is just 1.. .. Minus the top expression, x, as is.. Times the derivative of the bottom, which is 1 from the x and nothing from the -1, so it's just 1 there. And then it's all over the bottom squared.

the lower function squared. (x – 1) squared. Ok. So this is the derivative, but you should simplify it because things will cancel. Try simplifying within the top first.

within the numerator. This is x – 1 – x… Another x… All over (x – 1) ^ 2. Great news is that something cancels on top.

this becomes simpler. X – x cancels and you're just left with -1 on top. So, the simplest way to write your derivative f prime of x is.. -1 / (x – 1)^2 remember f prime just means the derivative. Alright.

second step. Again, plug in your x value, into the derivative f prime of x. What you can call that is f prime of 2, because you're plugging in 2. So the notation for that is.. F prime of 2 means you're putting 2 in place of every x in your derivative expression.

so in this case, that means -1 on top… And (2 – 1)^2 on the bottom. This simplifies to… -1 over 1^2, which is just 1. So your slope value…

your m is -1. I'll make that clear. Ok. Third step. Find the original y value if you don't already have it..

if you weren't given it. You weren't a full point, you were only half, just the x coordinate. So you have to find y. How do you do that? All you need to is take the original x, plug it in to your original equation, your original f… And find y, find the y value.

so you can call that f of 2 that will give you your y. F of 2 means you're plugging in 2 for x, so you have 2 / (2 – 1)… Which is 2 / 1, which is 2. Alright, so that is your y. So if you had to write a full point for the original point…

it would be 2 for x and 2 for y, or (2,2). That's important, you're going to use both of those numbers next. Because the last step is still to plug in your m, your y, and your x… All into the point-slope form for the equation of a line. And that point-slope form looks like…

y minus your y coordinate, your y1, which is 2… Equals m, your slope, which is -1… Times parentheses x, the x you want that in the equation. Otherwise it's not a full equation… Minus the x coordinate, or x1, which is also 2 in this problem.

alright, so you know, this is technically correct, you can leave your answer in point-slope form. Unless you have to rearrange it to get it into y = mx + b form, but technically this is correct so this is the equation for the tangent line to this graph at x = 2. Ok. One final problem. I'm giving you one that has a trig function in it.

f of x equals (sec x)^2 there's also a power in here, so this might look kind of scary, maybe… But just rely on what you know about derivatives, differentiating. In this case you're going to have to use the chain rule.. .. Yay chain rule.. Because there's a function inside of a power.

but even if this is a more complicated derivative you're still going to do the same steps. The first step is still… Find f prime of x. Find the derivative. F prime of x like i said, this is a power rule with another function inside, so…

use the chain rule because you don't just have x inside. You have some other x expression inside a power, you have to use the chain rule. First, you take the derivative of the power using the power rule. So the 2 power comes down out front as a coefficient, 2. You can still put parentheses.

sec x is still there… This power 2 is reduced by 1, so the power is 1. We don't need to write that. If we don't write it, it's assumed that the power is 1. You're not done.

you are not done, because it's not just x inside. Since there's a sec x, you also need to multiply by the derivative of sec x.. The inside function. You did the derivative of the outside function. Now you need to multiply by the derivative of the inside function.

what is the derivative of sec x? Well you can get that from a derivative table with the trig derivatives, common trig derivatives, or maybe you just know it. The derivative of sec x is… Sec x… Times tan x sec x tan x this is your f prime of x. Alright, second step.

plug in the x value that you're given. You're given pi over 3 for your x value. Plug it in to f prime. So you have f prime of pi over 3. It's a weird pi number for your x value..

but that's totally valid, especially for a trig function to have a pi/3x value that you're plugging in. So this would be 2 times sec(pi/3)… Times another sec(pi/3)… Tangent… Of pi/3.

alright, this can be simplified. This should come out to be just a number, a constant so you don't want to leave it like this. As for secant… It's always easier to find cosine first and then take the reciprocal, flip it. It's easier to evaluate secant if you first do cosine because secant is 1 / cos, so we have 1 / cos(pi/3)…

times another 1 / cos(pi/3)… Then you can just leave tan(pi/3). Unless you prefer to write sin over cos. But leave it as tan(pi/3). Now you need to use what you know about the unit circle…

which you may remember cos(pi/3)… Pi/3 is up here. Cos(pi/3) is the x coordinate of pi/3 which is one half. So this becomes 2 times… 1 over (1/2)…

times 1 over (1/2)… Times tan(pi/3). The tangent of pi/3… Is root 3. So it's times square root of 3.

these values – one half, one half, root 3 – you can also get them from your calculator if you're allowed to use a calculator. Just make sure that you're in radian mode before you plug in cos(pi/3). This simplifies because you have 2… 1 over (1/2) dividing by a fraction is the same as multiplying by its reciprocal so it's the same as multiplying by 2. So this is 2, and this is 2.

so that's 4. 2… Times 2. Four. And then you have a root 3 left over.

so your final f prime of pi/3, slope, value is 8 root 3. You can combine the 2 times 4 and that right there is your m, slope, value. Alright, that was step two. Step three, remember, you probably already have figured this out.. Step three is to find the y value, original y value if you're not given it.

but here you were given the full point, so y is 4. Your y1 in the point-slope form is 4. You have your x1… And you have your m, so now you're ready to plug into the point-slope form as your final step. And that is y minus y1, which is 4 equals m, which is 8 root 3, you can just write that.

you don't want to write the decimal version of that. 8 root 3 times x, you want x in your equation minus x1, which is pi/3. You can just write that, pi/3. And this right here is a totally valid way of writing your final answer. It's in point-slope form.

rearrange it if you have to, but this is a valid answer for the equation of the tangent line to this f of x, at that point. So i hope this video helped you figure out how to write the equation of a tangent line using derivatives. Look.. Anything that makes calculus easier is a good thing. So if this video helped you..

please click 'like' or subscribe! . Hi guys! I'm nancy.. And i'm going to show you how to find the equation of a tangent line using derivatives. What is a tangent line? Well, it just means if you have a graph… With some curve on it the tangent line is the line that touches your graph at one point exactly. It's a linear line that is tangent to your curve and sort of grazes it and touches at one point. So tangent line problems can come in many forms but thankfully you can use the same four steps in all of them so let me show you some examples.

ok. Say you have a problem that says to find the tangent line to y equals x cubed plus 2x at a point at the point: 1, 3. How do you start this problem? Well it's always gonna be the same four steps in these tangent line problems. The first step is to take the derivative of the equation that you're given. This y equals is also f of x.

it's also a function. You want to take the derivative of the equation you're given, so f prime of x is your first step. F prime of x and then you take the derivative of this expression. The derivative of x cubed… Remember for the power rule, you bring the power down and then decrease that power by 1.

so for the first term, we have 3x squared for the second term, the derivative of 2x is just 2. The x goes away. You have 3x squared plus 2. So that's your derivative, your f prime of x. The second step is always to take the x coordinate that you're given in the point…

and plug it in to f prime of x. You can call that f prime of 1. Since you're x coordinate is 1. So you take that number, and you're going to plug it in to what you just found, the derivative. And this is just notation for that, for plugging in 1.

anyway, when you plug in 1 for x, you have 3 times 1 squared plus 2… Which simplifies to 3 times 1 plus 2 which is just 5, so f prime of 1… Equals 5. This is going to be your slope number that you'll use. The third step is…

to find the y coordinate of your point, if you're not already given it so sometimes you're not given the y. In this problem, you're given the whole point, which could happen. So you have y equals 3, so you don't need to find y in this case. You can move on to the fourth step, the last step, is to plug in the numbers that you've found into the point-slope formula for a line. The point-slope formula…

in general looks like y minus y1 equals m times x minus x1. M is just your slope and that's going to be the f prime value that you found when you plugged in x so your m in this problem is 5. Which you found right here. X1 and y1 are the coordinates of the point that you found or that you were given. So this here you can also call x1, y1.

you have both of those numbers. Now you want an equation for the tangent line. That means that in your equation you need to leave an x variable and y variable in it. So it's ok to leave those there. In fact you want it.

you can go ahead and you can write your equation by plugging in values for m, for x1, and for y1. So your x1 is the coordinate 1. Your y1 is the y coordinate of the point. The original point, which is 3. So if you rewrite this, your equation is y – 3 equals the slope 5, times (x – 1).

notice you're leaving x and y in the equation because you want it to be an equation for y in terms of x. So this… This is correct. This is a correct answer. Sometimes it's considered simpler to move it around and change it to be in slope intercept form or y = mx + b form.

so you can do that with some simple algebra if you were to multiply, distribute out, the… 5x – 5, and rearrange the numbers you could get into y equals form. If you were to multiply, distribute out, the… 5x – 5, and rearrange the numbers you could get into y equals form. Y = 5x – 5 + 3… Which is just minus 2.

so this.. Y = 5x – 2 is your equation of the tangent line to this f of x at that point. Alright, say you have f(x) = x / (x – 1) and you have to find the equation of the tangent line at x = 2. This looks a lot more complicated but the truth is it's the same four steps no matter how complicated your f(x), your equation gets. But the truth is it's the same four steps no matter how complicated your f(x), your equation gets. This could be ridiculous..

but as long as you know how to take f prime of x, the derivative, you can still do all the same steps and get the equation of the linear tangent line. So the first step is still, find f of x. Find the derivative of your given equation. You want f prime of x.. Now since your equation is x / (x – 1) this is a fraction.

this is a quotient. So you'll have to use the quotient rule to take the derivative. Which is a little more complicated than you might like.. But the steps are pretty straightforward. Think back to doing the quotient rule..

it's the bottom of the function times the derivative of the top.. Derivative of x is just 1.. .. Minus the top expression, x, as is.. Times the derivative of the bottom, which is 1 from the x and nothing from the -1, so it's just 1 there. And then it's all over the bottom squared.

the lower function squared. (x – 1) squared. Ok. So this is the derivative, but you should simplify it because things will cancel. Try simplifying within the top first.

within the numerator. This is x – 1 – x… Another x… All over (x – 1) ^ 2. Great news is that something cancels on top.

this becomes simpler. X – x cancels and you're just left with -1 on top. So, the simplest way to write your derivative f prime of x is.. -1 / (x – 1)^2 remember f prime just means the derivative. Alright.

second step. Again, plug in your x value, into the derivative f prime of x. What you can call that is f prime of 2, because you're plugging in 2. So the notation for that is.. F prime of 2 means you're putting 2 in place of every x in your derivative expression.

so in this case, that means -1 on top… And (2 – 1)^2 on the bottom. This simplifies to… -1 over 1^2, which is just 1. So your slope value…

your m is -1. I'll make that clear. Ok. Third step. Find the original y value if you don't already have it..

if you weren't given it. You weren't a full point, you were only half, just the x coordinate. So you have to find y. How do you do that? All you need to is take the original x, plug it in to your original equation, your original f… And find y, find the y value.

so you can call that f of 2 that will give you your y. F of 2 means you're plugging in 2 for x, so you have 2 / (2 – 1)… Which is 2 / 1, which is 2. Alright, so that is your y. So if you had to write a full point for the original point…

it would be 2 for x and 2 for y, or (2,2). That's important, you're going to use both of those numbers next. Because the last step is still to plug in your m, your y, and your x… All into the point-slope form for the equation of a line. And that point-slope form looks like…

y minus your y coordinate, your y1, which is 2… Equals m, your slope, which is -1… Times parentheses x, the x you want that in the equation. Otherwise it's not a full equation… Minus the x coordinate, or x1, which is also 2 in this problem.

alright, so you know, this is technically correct, you can leave your answer in point-slope form. Unless you have to rearrange it to get it into y = mx + b form, but technically this is correct so this is the equation for the tangent line to this graph at x = 2. Ok. One final problem. I'm giving you one that has a trig function in it.

f of x equals (sec x)^2 there's also a power in here, so this might look kind of scary, maybe… But just rely on what you know about derivatives, differentiating. In this case you're going to have to use the chain rule.. .. Yay chain rule.. Because there's a function inside of a power.

but even if this is a more complicated derivative you're still going to do the same steps. The first step is still… Find f prime of x. Find the derivative. F prime of x like i said, this is a power rule with another function inside, so…

use the chain rule because you don't just have x inside. You have some other x expression inside a power, you have to use the chain rule. First, you take the derivative of the power using the power rule. So the 2 power comes down out front as a coefficient, 2. You can still put parentheses.

sec x is still there… This power 2 is reduced by 1, so the power is 1. We don't need to write that. If we don't write it, it's assumed that the power is 1. You're not done.

you are not done, because it's not just x inside. Since there's a sec x, you also need to multiply by the derivative of sec x.. The inside function. You did the derivative of the outside function. Now you need to multiply by the derivative of the inside function.

what is the derivative of sec x? Well you can get that from a derivative table with the trig derivatives, common trig derivatives, or maybe you just know it. The derivative of sec x is… Sec x… Times tan x sec x tan x this is your f prime of x. Alright, second step.

plug in the x value that you're given. You're given pi over 3 for your x value. Plug it in to f prime. So you have f prime of pi over 3. It's a weird pi number for your x value..

but that's totally valid, especially for a trig function to have a pi/3x value that you're plugging in. So this would be 2 times sec(pi/3)… Times another sec(pi/3)… Tangent… Of pi/3.

alright, this can be simplified. This should come out to be just a number, a constant so you don't want to leave it like this. As for secant… It's always easier to find cosine first and then take the reciprocal, flip it. It's easier to evaluate secant if you first do cosine because secant is 1 / cos, so we have 1 / cos(pi/3)…

times another 1 / cos(pi/3)… Then you can just leave tan(pi/3). Unless you prefer to write sin over cos. But leave it as tan(pi/3). Now you need to use what you know about the unit circle…

which you may remember cos(pi/3)… Pi/3 is up here. Cos(pi/3) is the x coordinate of pi/3 which is one half. So this becomes 2 times… 1 over (1/2)…

times 1 over (1/2)… Times tan(pi/3). The tangent of pi/3… Is root 3. So it's times square root of 3.

these values – one half, one half, root 3 – you can also get them from your calculator if you're allowed to use a calculator. Just make sure that you're in radian mode before you plug in cos(pi/3). This simplifies because you have 2… 1 over (1/2) dividing by a fraction is the same as multiplying by its reciprocal so it's the same as multiplying by 2. So this is 2, and this is 2.

so that's 4. 2… Times 2. Four. And then you have a root 3 left over.

so your final f prime of pi/3, slope, value is 8 root 3. You can combine the 2 times 4 and that right there is your m, slope, value. Alright, that was step two. Step three, remember, you probably already have figured this out.. Step three is to find the y value, original y value if you're not given it.

but here you were given the full point, so y is 4. Your y1 in the point-slope form is 4. You have your x1… And you have your m, so now you're ready to plug into the point-slope form as your final step. And that is y minus y1, which is 4 equals m, which is 8 root 3, you can just write that.

you don't want to write the decimal version of that. 8 root 3 times x, you want x in your equation minus x1, which is pi/3. You can just write that, pi/3. And this right here is a totally valid way of writing your final answer. It's in point-slope form.

rearrange it if you have to, but this is a valid answer for the equation of the tangent line to this f of x, at that point. So i hope this video helped you figure out how to write the equation of a tangent line using derivatives. Look.. Anything that makes calculus easier is a good thing. So if this video helped you..

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