Main content

## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition) > Unit 2

Lesson 20: Chain rule- Chain rule
- Worked example: Derivative of cos³(x) using the chain rule
- Worked example: Derivative of ln(√x) using the chain rule
- Worked example: Derivative of √(3x²-x) using the chain rule
- Chain rule intro
- Chain rule overview
- Worked example: Chain rule with table
- Chain rule with tables
- Quotient rule from product & chain rules
- Chain rule with the power rule
- Applying the chain rule graphically 1 (old)
- Applying the chain rule graphically 2 (old)
- Applying the chain rule graphically 3 (old)
- Chain rule

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Applying the chain rule graphically 2 (old)

Sal solves an old problem where the graph of a functions g is given, and he evaluate the derivative of [g(x)]³ at a point. Created by Sal Khan.

## Want to join the conversation?

- why is the derivative of g(x)^3 not just 3(g(x))^2 like the power rule says? Why do you have to multiply by g'(x)?(22 votes)
- When one function applies to the output of another, the inner function produces a rate of change and the outer one applies to the output, multiplying that rate of change. If you simply apply the power rule in this case, you take into account the change produced by the outer function but not the change produced by the inner function.

It might be easier to see this if you work an example using x^2 as the inner function g(x), so we have (x^2)^3 (x squared, quantity to the third power). We know this composite function is equal to x^6, so the derivative has to be 6x^5. If you take the derivative only "on the outside" you get 3(x^2)^2, which is 3x^4, plainly incorrect. Using the chain rule, though, we multiply this "outer derivative" by the "inner derivative," which is 2x, and voila, we get the correct result, 6x^5.(39 votes)

- at3:31, you say it is -4 over 2 for the slope. It is confusing because you use the marks on the graph instead of the actual numbers. It's still the same though, right?(6 votes)
- Yes. The scale is 2, so the slope is -2/1 = -2(0 votes)

- i have a few questions, i'm in my last year of highschool in the netherlands now and never learned about sec. where do i find it on my TI 84 calculator and what is it exactly? and were do i find the extra symbols input thingy? i tried the exercise for this video but couldnt find a way to type -1 2/3.(0 votes)
- Secant is the reciprocal of cosine. Cosecant is the reciprocal of sine. Cotangent is the reciprocal of tangent.(3 votes)

- This is a pretty basic question, but when you are using the chain rule, how do you tell which is the "inside function" and which is the "outside function?" Sometimes there are multiple ways to divide a composite function, aren't there?(2 votes)
- i dont understand how to find the slop value?why i increase 2 or decrease 4 ,rather how do i get this?(1 vote)
- He actually has the tangent line already drawn for him.

The slope of the tangent line is change of y/change of x

Change of x is = 5-4

Change of y is = 1-3 Therefore the slope of the tangent line is equal to -2/1(2 votes)

- at2:44,Sal used 2 points on the tangent line to calculate the slope, but wouldn't it be easier to just use the value of the tangent line at point (4,3) to do that, i mean the it's a straight line so the rate of change at any point on the tangent line should be the same right?(1 vote)
- The value of the tangent line at point (4,3) is equal to the value of the function at that point. We were looking for the slope of the tangent line.(1 vote)

- At3:08, the run was 2 and rise was -4. But the actual run was 5 - 4 = 1 And the actual rise was 1 - 3 = -2. Although the slope was correct, but the method wasn't.(1 vote)
- How do you check if the answer is correct?(2 votes)
- If you mean on the Khan academy exercises, look to the left and look for the little green box that says "check answer"(0 votes)

- How is g'(4) = slope of the tangent a x=4?(1 vote)
- shouldn't g'(x) equal the equation of the tangent line, not just the slope of the tangent line?

the equation of the tangent line is g'(x)= -2x+11, not g'(x) = 2.(1 vote)

## Video transcript

Given capital F of x is equal
to g of x to the third power where the graph of g and its
tangent line at x equals 4 are shown, what is the
value of F prime of 4? So they give us g of x
right over here in blue. And they show us
the tangent line at x equals 4 right over here. So we need to figure
out F prime of 4. So let's just rewrite this
information they've given us. We know that F of x is equal
to g of x to the third power. So I'll write it like this,
g of x to the third power. So we want to figure
out what F prime of x is when x is equal to 4. So let's just take
the derivative here of both sides
with respect to x. So take the derivative
of the left-hand side with respect to x, and
take the derivative of the right-hand side
with respect to x. So the left-hand
side, this is just going to be capital
F prime of x. Now on the right-hand
side I have a composite. I have g of x to
the third power. So first we can view this as the
product of the derivative of g of x to the third power
with respect to g of x. So there we could
literally just apply what we know about
the power rule. The derivative of x to the
third with respect to x is 3x squared. So the derivative of gx to the
third with respect to g of x is just going to be three times
g of x to the second power. And then we're going to multiply
that times the derivative of g of x with respect to x. So times g prime of x And this comes straight
out of the chain rule. Derivative of this,
the derivative of g of x to the third
with respect to g of x, which is this,
times the derivative of g of x with respect to x,
which is that right over there. So now let's just substitute. We want to figure out
what this derivative is when x is equal to 4. So we could say
that F prime of 4 is equal to 3 times g of 4
squared times g prime of 4. So what is g of 4 going to be? Well, we can just look at
our function right over here. When x equals 4, our
function is equal to 3. So g of 4 is equal to 3. And what's g prime of 4? So when x equals
4, g prime of 4 is the slope of the tangent line. And they've drawn the tangent
line when x equals 4 here. So what is the
slope of this line? So we just have to think about
change in y over change in x. And I'll look at that between
two integer-valued coordinates. So it looks like between
these two points. And when we increase x
by 2, we decrease y by 4. So as you remember,
slope is rise over run, or change
in y over change in x. So the slope of the tangent
line here, the slope is equal to our
change in y negative 4 over our change in x. And this is going to
be equal to negative 2. So this simplifies to F
prime of 4 is equal to-- I'll do this in a new color--
3 squared is 9 times 3 is 27 times negative 2, which
is equal to negative 54. So F prime of 4 is negative 54.