Find the derivative of f(x) = sinx*cosx using the product rule

Expert Answers
hala718 eNotes educator| Certified Educator

Given f(x) = sinx*cosx

We need to find f'(x) using the product rule.

Then we will assume that f(x) = u*v such that:

u= sinx ==> u' = cosx

v= cosx ==> v' = -sinx

Then we know that:

f'(x) = u'v + uv'

==> f'(x)= cos^2 x - sin^2x

But we know that cos2x = cos^2x - sin^2 x

==> f'(x) = cos^2 x - sin^2 x = cos2x

justaguide eNotes educator| Certified Educator

Using the product rule for f(x) = g(x)*h(x), f'(x) = g'(x)*h(x) + g(x)*h'(x)

Here f(x) = (sin x)*(cos x)

f'(x) = (sin x)'*(cos x) + (sin x)*(cos x)'

=> f'(x) = (cos x)^2 - (sin x)^2

=> f(x) = cos 2x

The derivative of f(x) = (sin x)*(cos x) is cos 2x

Access hundreds of thousands of answers with a free trial.

Start Free Trial
Ask a Question