1. If f'(1/3*x^3)=2x^5, what is f'(x)?

2. Find the limit lim n-infinity (1/(2n+1)+1/(2n+2)+...+1/(2n+n)
by computing a suitable definite integral.

I am stressed out, thank you a lot for you help :) x

Question

1.  If f'(1/3*x^3)=2x^5, what is f'(x)?

2. Find the limit lim n->infinity (1/(2n+1)+1/(2n+2)+...+1/(2n+n) 
by computing a suitable definite integral.
                         
I am stressed out, thank you a lot for you help :) x

turingtest · Answer

for the first one, integrate 2x^5 to get f(1/3*x^3), from which f(x) should  be obvious
from there getting f'(x) is easy

anonymous · Answer

@TuringTest Not so fast! ;P

$\large\displaystyle \int f^{\prime}\left(\frac{x^3}{3}\right)\,dx \neq f\left(\frac{x^3}{3}\right)$.

You need to take into consideration the $\large \frac{1}{3}x^3$ term inside $\large f^{\prime}$.  Therefore, you have to make a substitution first before integrating.