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So now we're going to take the derivative of the sine of

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the natural log of x squared, so now we have a function

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thats the composite of a function thats a composite of another function

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So one way you could think of it: if you set f(x)

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equal to sine of x and g of x

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g of x being the natural log of x

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the natural log of x, and lets say

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h of x equaling x squared

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then this thing right here is the exact same thing as

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take the derivative with respect to x

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f of g of h of x

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and what I want to do is kind of think of how I want to do it in my head

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not having to write all of the chain rule notation, so the way I would think

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about this if I were doing this in my head is the derivative of this outer function of f with respect to the level of composition directly below it

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so the derivative of sine of x is consine of x

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but instead of it being cosine of x it's going to be cosine of whatever was inside of here, so it's going to be cosine of

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natural log, cosine of natural log of x sqaured

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and so you could really view this, this part that I just

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wrote right over here as f prime of g of h of x

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if you want to keep track of things

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Just took the derivative of the outer with respect to whatever was inside of it

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have to take the derivative of the inside with respect to x, but now we have

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another composite function, so we're going to multiply this times-we're going to do the chain rule

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again, the derivative of, we're going to take the derivative of

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ln with respect to x squared

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so the derivative of ln x is 1 over x, but now we have