The Chain Rule: Solutions

Let

so that

At this point, there is no further convenient simplification.

With u(x)=2x²-3x+1,

Here, the chain rule is used along with the product rule to find

Those wishing to be clever may recognize (see Trig Identities) that

and that the derivative of this form is the same as that obtained from the chain rule.

Repeated application of the chain rule gives

and this can be shown to reduce to

As a check, the original expression for S(r) may be shown to reduce to

and the quotient rule gives the above form more directly.

Letting u(x)=cos(x),