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

The following table lists the values of functions F and H and of their derivatives f prime and H prime for x is equal to 3. So, let's just tell us when x is equal to three, the value of the function is six. F of three is six. You could view it that way: H of three is zero. F prime of three is six and H prime of three is four.

And now they want us to evaluate the derivative with respect to x of the product of F of x and H of x when x is equal to 3. So, one way you could view this is if we viewed some function G of x as being equal to the product of F of x and H of x. This expression is the derivative of G of x. So, we could write G prime of x is equal to the derivative with respect to x of F of x times H of x, which is what we see right here and which is what we want to evaluate at x equal to 3.

So, we essentially want to evaluate G prime of three. This is what they're asking us to do. Well, to do that, let's go first up here; let's just think about what this is doing. They're asking us to take the derivative with respect to x of the product of two functions that we have some information about. Well, if we're taking the derivative of the product of two functions, you could imagine that the product rule could prove useful here.

So, I'm just going to restate the product rule. This is going to be equal to the derivative of the first function, F prime of x, times the second function—not taking its derivative—plus the first function, not taking its derivative, F of x, times the derivative of the second function, H prime of x.

So, if you're trying to find G prime of 3, well that's just going to be F prime of 3 times H of 3 plus F of 3 times H prime of three. And lucky for us, they give us what all of these things evaluate to. F prime of three right over here, they tell us F prime when x is equal to 3 is equal to 6. So this right over here is six.

H of three, they give us that too. H of three when x is three, the value of our function is zero. So this is zero. So this first term, you just get six times zero, which is going to be zero. But we'll get to that now. F of three. F of three, well the function when x is equal to 3, or the function F of three, is equal to 6, so that is six.

And then finally, the H prime evaluated at three: H prime of x when x is equal to 3, H prime of x is equal to 4, or you could say this is H prime of three. So this is four. And so, there you have it; this is going to evaluate to 6 times 0, which that’s all just going to be 0.

0 plus 6 times 4, which is going to be equal to 24, and we're done.