Question

How do you find the first derivative for `f(x)= (sin x)(cos x)`?

Answer 1

Using the multiplication rule for derivatives, which states that the derivative of a product `(f*g)'` equals the following:

`(f*g)'=f'*g+f*g'`

Since the derivative of `sin(x)` is `cos(x)`, and the derivative of `cos(x)` is `-sin(x)`, we have that

`(sin(x) * cos(x))' = (sin(x))' * cos(x) + sin(x) * (cos(x))'`
`= cos(x)*cos(x) + sin(x) (-sin(x))=`
`= cos^2(x) - sin^2(x)`

This would of course be a perfect answer, but I think that noticing that `cos^2(x) - sin^2(x)=cos(2x)` would be more elegant.