The chain rule is probably the most used rule for
differentiating. Here we will use what we called version 1, which
says that
(f(g(x)))′=f′(g(x))⋅g′(x)
In particular, (f(◻))′=f′(◻)⋅◻′. Note: the "inside" function here is represented by
g(x)=◻; it is important to be able to find the inside part,
as you will see in the next example.
Example: Compute the derivative of y=sin(x2).
DO: Try this problem
before looking at the solution, using the box above. Here,
f(x) is sin(x) and g(x) is x2, so that
f(g(x))=sin(x2). Hint: write down sin(x2) --
what is the inside part? Put your finger over the inside
function, differentiate the outside function (leaving the
inside part alone) then multiply by the derivative of the inside
part. Try it.
Solution: Since the derivative of x2 is 2x and the
derivative of sine is cosine, ddx(sin(x2))=cos(x2)⋅(x2)′=cos(x2)⋅2x.