Chain Rule: The General Logarithm Rule - Concept

Talking about the chain rule and in a moment I'm going to talk about how to differentiate a special class of functions where they're compositions of functions but the outside function is the natural log. But first I want to take a look at an identity that comes from a property of natural logs e to the natural log of x equals x, now if I differentiate both sides this equation I'll get a surprising and useful result. So I'm differentiating the left side e to the lnx and I'm differentiating the right side right. If 2 functions are equal for all values of x then their derivative should be equal. So on the right side you can see that the derivative with respect to x of x is 1 right this is just a linear function derivative as a slope and on the left side I can use the chain rule.
The derivative e to the lnx is going to be e to the lnx times the derivative of lnx, let's pretend we don't know that, we don't know the derivative of lnx so times the derivative of lnx and then we can divide both sides by e to the lnx and we get the derivative with respect to x of lnx equals 1 over e to the lnx. Now let's recall that e to the lnx is just x so this is the same as 1 over x what we just proved is that the derivative with respect to x of lnx is 1 over x. Now we've been using this result for a while but I don't think we proved it so here's an actual proof of that derivative result.
Now let's go on the chain rule, so you recall the chain rule tells us how the derivative differentiate a composite function and for composite functions there's an inside function and an outside function and I've been calling the inside function g of x and the outside function f of x. Well in this case we're going to be dealing with composite functions with the outside functions natural log. So if you're differentiating the natural log of g of x the derivative is 1 over g of x times g prime of x and this is just a special case of the chain.