Worked example: Product rule with mixed implicit & explicit | AP Calculus AB | Khan Academy

Let F be a function such that F of negative 1 is 3 and F prime of negative 1 is equal to 5. Let G be the function G of X is equal to 1 over X. Let capital F function to find it as the product of those other two functions.

What is capital F prime of negative 1? Well, we can just apply the product rule here. Let me just rewrite, let me just essentially state the product rule. Capital F prime of X is going to be equal to, since capital F of X is the product of these two functions, we apply the product rule. This is going to be F prime of X times G of X plus f of X times G prime of X.

And so, if we want to evaluate this at F at negative 1, capital F prime at negative 1 is equal to F prime of negative 1 times G of negative 1 plus function f evaluated at negative 1 times the derivative of G evaluated at negative 1. Now, let's see if we can figure these things out.

So, do they tell us this anywhere? Can we figure this out? F prime of negative 1, well, they tell us, right? If your F prime of negative 1 is equal to 5, so this is equal to 5. Now, let's actually stick with F. What is F of negative 1? Well, they tell that to us right over here. F of negative 1 is equal to 3, so F of negative 1 is equal to 3.

Now G, G of negative 1 and G prime of negative 1, they don't give it to us explicitly here, but we could figure it out. We can, we know that, well, if G of X is equal to this, G of negative 1 is equal to 1 over negative 1, which is equal to negative 1, so this is equal to negative 1.

And then last but not least, if we want to find G prime of negative 1, we just have to take the derivative of this. So G prime of X, actually let me just rewrite G of X. G of X, 1 over X, is just the same thing as X to the negative 1, so we're going to use the power rule to figure out G prime of X is equal to bring that exponent out front, negative one times X to the, and then decrement the exponent negative two power.

So G prime of negative 1 is equal to negative one times negative one to the negative two power, and that's just the same thing as negative one over negative one squared. This is one, so this is just all going to evaluate to negative one. So this is negative one, and so we have five times negative one, which is negative 5 plus three times negative one, which is negative three, which is equal to negative eight.

So f prime of negative one is equal to negative eight.