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

1 Answer
Feb 21, 2017

Answer:

#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)#