Question

How do you use the product rule to differentiate `f(x)=e^x(sqrtx+5x^3)`?

Answer 1

`f'(x) = e^x(sqrtx+5x^3)+e^x(1/(2sqrtx)+15x^2)`

Explanation:

The function `f(x)=e^x(sqrtx+5x^3)` can be written as `f(x)=g(x).h(x)` so we can use the product rule to find its derivative.

In this case `g(x)=e^x` and `h(x)=sqrtx+5x^3`.

We differentiate `g(x)=e^x` to get `g'(x)=e^x`

And `h(x)=sqrtx+5x^3=x^(1/2)+5x^3` to get `h'(x)=1/2x^(-1/2)+15x^2=1/(2sqrtx)+15x^2`

The product rule states that `f'(x)=g'(x).h(x)+h'(x).g(x)`

Putting our functions into that equation gives `f'(x) = e^x(sqrtx+5x^3)+e^x(1/(2sqrtx)+15x^2)`