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Applying the chain rule graphically 2 (old)

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.