Question

How do you differentiate `f(x)=(cosx+x)(-x^2-e^x)` using the product rule?

Answer 1

`f'(x) = x^2sin x -3x^2 - 2xcosx + e^x(sin x-cosx - 1 - x)`

Explanation:

The product rule states that
`d/(dx)(ab) = b (da)/(dx) + a (db)/(dx)`
Therefore,
`f'(x) = (-x^2 - e^x)d/dx(cos x + x) + (cosx + x)d/dx(-x^2 - e^x)`
`= (-x^2 - e^x)(-sin x + 1) + (cosx + x)(-2x - e^x)`
`= x^2sin x -3x^2 + e^xsin x - e^x - 2xcosx - e^x cosx - xe^x`

Which doesn't really clean up any more.