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

1 Answer
Feb 21, 2017

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)