Question

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

Answer 1

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

Explanation:

The product rule for two functions is

`(g(x)h(x))^'=g^'(x)h(x)+g(x)h^'(x)`

In this case, let `g(x)=e^x+x` and `h(x)=x^2-x`. Taking derivatives gives `g^'(x)=e^x+1` and `h^'(x)=2x-1`. Now plugging these expressions into the formula gives

`f^'(x)=(e^x+1)(x^2-x)+(e^x+x)(2x-1)`

Now distribute and simplify

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

`f^'(x)=3x^2-2x+x^2e^x+xe^x-e^x`

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