Question

How do you differentiate `f(x)=e^x*sin^2x` using the product rule?

Answer 1

`color(blue)(f'(x) = e^xsinx(sinx + 2cosx)`

Explanation:

We're asked to find the derivative

`d/(dx) [e^x*sin^2x]`

Using the power rule*, which is

`d/(dx) [uv] = v(du)/(dx) + u(dv)/(dx)`

where

• `u = e^x`

• `v = sin^2x`:

`f'(x) = sin^2xd/(dx)[e^x] + e^xd/(dx)[sin^2x]`

The derivative of `e^x` is `e^x`:

`f'(x) = e^xsin^2x + e^xd/(dx)[sin^2x]`

To differentiate the `sin^2x` term, we'll use the chain rule:

`d/(dx) [sin^2x] = d/(du) [u^2] (du)/(dx)`

where

• `u = sinx`

• `d/(du) [u^2] = 2u` (from power rule):

`f'(x) = e^xsin^2x + e^x*2sinxd/(dx)[sinx]`

The derivative of `sinx` is `cosx`:

`f'(x) = e^xsin^2x + 2e^xsinxcosx`

Or

`color(blue)(ulbar(|stackrel(" ")(" "f'(x) = e^xsinx(sinx + 2cosx)" ")|)`