 In a kind and gentle universe, we'd always be given functions we could differentiate. We don't live in that universe. But we might still need to find derivatives. So one possibility is we might have a table of values of a function and its derivatives. So we might have such information and we might want to find the derivative of a composite function. And we can still apply the chain rule. So remember the derivative of f of g of x is the derivative of f evaluated at g of x times the derivative of g of x. And so we can write down our derivative. Now we want to find the derivative at x equal to 3, so remember equals means replaceable. Every place we see an x will replace it with a 3. Well that doesn't do us much good. We don't know what g of 3 is. If only there was some way we could determine the value of g of 3. Oh wait, it's right here in the table. g of 3 is equal to 1. If only there was some way we could determine the value of f prime of 1 and g prime of 3. And they're right here. f prime of 1 is equal to 2. And g prime of 3 is also equal to 2. Which gives us our derivative. And we can apply this approach to any problem that involves the chain rule. So suppose I want to find the derivative of f of x cubed. So remember an easy way to apply the chain rule is to ignore everything but the outermost function. So this is the derivative of something cubed. And the derivative of something cubed will be. And remember you can never go wrong by applying the chain rule. So we should make sure that we have this derivative of whatever was inside the parentheses. And put things back where you found them. Inside our grouping symbol was f of x. And so that should go inside all of our grouping symbols. But wait, there's more. Here I want the derivative of f of x. And I might not know what it is but at least I can write it. That's f prime of x. Now I want to know this value at x equals 1. It's replaceable so I'll replace. And our table gives us the values of f of 1 and f prime of 1. So we'll use those values. Which gives us our derivative. And we can even do more complicated functions this way. So if I want to find the derivative. It helps to think about this as something to power negative 1. Ignore everything but the outermost function. Thing to power minus 1 has derivative. Again, don't forget the chain rule. Put things back where you found them. We still have one more derivative to evaluate. So we'll differentiate. And we'll rewrite this slightly to make it easier to evaluate. And at x equals 5 we know these values. Which will give us the derivative.