Question

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

Answer 1

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

Explanation:

We have: `f(x) = (x^(2) + sin(x))(e^(x) - cos(x))`

`Rightarrow f'(x) = (x^(2) + sin(x)) cdot frac(d)(dx)(e^(x) - cos(x)) + (e^(x) - cos(x)) cdot frac(d)(dx)(x^(2) + sin(x))`

`Rightarrow f'(x) = (x^(2) + sin(x))(e^(x) + sin(x)) + (e^(x) - cos(x))(2 x + cos(x))`

`Rightarrow f'(x) = x^(2) e^(x) + x^(2) sin(x) + sin(x) e^(x) + sin^(2)(x) + 2 x e^(x) + cos(x) e^(x) - 2 x cos(x) - cos^(2)(x)`