Find the derivative of the function

*f*(*x*) = *x*^{3}sin *x *cos *x*

in two different ways. Give the answer with everything multiplied out (instead of factoring out common factors).

Answer

One way is to think of the function as

*f*(*x*) = (*x*^{3}sin *x*)cos *x*.

In this case,

*f'*(*x*) = (*x*^{3}sin *x*)'cos *x* + (*x*^{3}sin *x*)(cos *x*)'.

We need to use the product rule again to find one of the derivatives we need:

Now we put this back into the product rule:

The other way is to think of the function as

*f*(*x*) = (*x*^{3})(sin *x *cos *x*).

In this case, we'll need to use the product rule to find

(sin *x* cos *x*)'

before we can find the derivative of the original function. Here we go:

While cos^{2}*x* + sin^{2}*x* = 1, cos^{2}*x*-sin^{2}*x* isn't anything in particular, we can't make this any nicer.Now we can use the product rule to find *f*'.

Thankfully, we find the same answer either way.