The chain rule is probably the most used and abused rule for differentiating.
Here we will use Version 1, which says that
(f(g(x)))′=f′(g(x))⋅g′(x)
In particular, (f(◻))′=f′(◻)⋅◻′, and we can imagine to put whatever other function inside the box. (What happens if we put x in it?)
Example: Compute the derivative of y=sin(x2).
Solution:
We take g(x)=x2
and f(x)=sin(x), so that f(g(x))=sin(x2).
Since the derivative of x2 is 2x and the derivative of
sine is cosine,
ddx(sin(x2))=cos(x2)⋅(x2)′=cos(x2)⋅2x.