Question

How do you differentiate `f(x)= (x^3+2x+1)(2+x^-2)` using the product rule?

Answer 1

`f'(x)=6x^2-x^-2-2x^-3-2`

Explanation:

The product rule is `d/dx(uv)=u (dv)/dx+v (du)/dx`; so
`f'(x)=(x^3+2x+1)d/dx(2+x^-2)+d/dx(x^3+2x+1)(2+x^-2)`

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

`f'(x)=(-2)(x^3+2x+1)/(x^3)+(6x^2+3x^-2+4+2x^-2)`

`f'(x)=(-2)(1+2x^-2+x^-3)+(6x^2+4+5x^-2)`

`f'(x)=-2-4x^-2-2x^-3+6x^2+4+5x^-2`

`f'(x)=6x^2-x^-2-2x^-3-2`