Derivative of sin(ln(x²))

so now we're going to attempt to take the derivative of the sine of the natural log of x squared so now we have a function that's the composite of a function that's a composite of another function so one way you can think of it if you set f of X f of X is being equal to sine of X and G of X G of X being the natural log of X the natural log of X and that's the F G let's say H of X H of X equaling x squared then this thing right over here is the exact same thing as trying to take the derivative with respect to X of F of F of G of F of G of H of X of H of X and what I want to do is kind of do think about it how I would do it in my head without having to write all the chain rule notation so the way I would think about this if I were doing this my head is the derivative of this outer function of f with respect to the the level of composition directly below it so the derivative of sine of X is cosine of X but instead of it being cosine of X it's going to be cosine of cosine of whatever was inside of here so it's going to be cosine of natural log let me write that in that same color cosine of natural log of x squared x squared don't do X in that same yellow color cosine of x squared x squared and so you could really view this this part what I just write over here as as f prime that this is F prime of G of H of X this is f prime of G of H of X G of H of X if you want to keep track of things so I just took the derivative of the outer with respect to whatever was inside of it and now I have to take the derivative of the inside with respect to X but now we have another composite function so we're going to multiply this times we're going to the chain rule again the derivative of we're going to take the derivative of Ln with respect to x squared so the derivative of Ln of X is 1 over X but now we're going to have 1 over not X but 1 over x squared 1 over x squared so to be clear this part right over here this part right over here is G prime of not X if it was G prime of X this would be 1 over X but instead of an X we have our H of X there we have our x squared so it's G prime of x squared and then finally we can take the derivative of our of our inner function let me write so we could write this as G prime of H of X G prime of H of X and finally we just have to take the derivative of our innermost function with respect to X so the derivative of 2x with respect to the derivative of x squared with respect to X is 2x so times H prime of X let me make everything clear so this let me see so what we have right over here in purple this this and this are the same things 1 expressed concretely 1 expressed abstractly this this and this are the same thing expressed concretely and abstractly and then finally this and this are the same thing expressed concretely and abstractly but then we're done all we have to do is simple to be done is to just simplify this so if we just change the order in which we're multiplying we have 2x over x squared so I can cancel some out so this x over X 2x over x squared is the same thing as 2 over X and we're multiplying it times all of this business so we're left with 2 over X this goes away 2 over x times the cosine of the natural log of x squared so it seemed like a very daunting derivative but we just say ok what's the derivative of sine of something with respect to that's something well that's cosine of that something and then we go in one layer what's the derivative of that something well and that's something we have another composition so the derivative of Ln of X or Ln of something with respect to another something well that's going to be one over the something so we had gotten a one over x squared here that that that squared got canceled out and then find the derivative of this inner most function it's like peeling an onion the derivative of this inner function with respect to X which was just 2x which we got right over here this was 1 over x squared this was 2x before we did any cancelling out so hopefully that helps clear things up a little bit