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Worked example: Chain rule with table | Derivative rules | AP Calculus AB | Khan Academy

2m read
·Nov 11, 2024

The following table lists the values of functions f and g and of their derivatives f prime and g prime for the x values negative 2 and 4.

And so, you can see for x equals negative 2, x equals 4, they give us the values of f, g, f prime, and g prime. Let function capital F be defined as the composition of f and g, it's lower case f of g of x. They want us to evaluate f prime of 4.

So, you might immediately recognize that if I have a function that can be viewed as the composition of other functions, that the chain rule will apply here. And so, I'm just going to restate the chain rule. The derivative of capital F is going to be the derivative of lower case f, the outside function, with respect to the inside function, so lower case f prime of g of x times the derivative of the inside function with respect to x, times g prime of x.

And if we're looking for f prime of 4, f prime of 4, well everywhere we see an x, we replace it with a 4. That's going to be lower case f prime of g of 4 times g prime of 4. Now, how do we figure this out? They haven't given us explicitly the values of the functions for all x's, but they've given it to us at some interesting points.

So, the first thing you might want to figure out is, well, what is g of 4 going to be? Well, they tell us when x is equal to 4, g of 4 is negative 2. This tells us that the value that g of x takes on when x is equal to 4 is negative 2. So this right over here is negative 2.

And so, this first part is f prime of negative 2. So what is f prime? What is f prime of negative 2? Well, when x is equal to negative 2, f prime is equal to one. So this right over here is f prime of negative two; that is equal to one.

And now, we just have to figure out what g prime of four is. Well, when—let me circle this—g prime of 4, when x is equal to 4, and I'll scroll down a little bit, when x is equal to 4, g prime takes on the value 8.

So, there you have it. f prime of 4 is equal to 1 times 8, which is equal to eight, and we're done.

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