Find the derivative of the function
f(x) = x3sin x cos x
in two different ways. Give the answer with everything multiplied out (instead of factoring out common factors).
One way is to think of the function as
f(x) = (x3sin x)cos x.
In this case,
f'(x) = (x3sin x)'cos x + (x3sin 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) = (x3)(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 cos2x + sin2x = 1, cos2x-sin2x 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.