The derivative of a function raised to the nth power

Question

I just started learning about derivatives and I am pretty familiar with almost all the concepts related to derivatives but there has been something that I yet have not fully mastered and that is finding the derivative of a function raised to the nth power where n is a rational number. And also can you please show the proof? P.S: I already know that the derivative of x raised to the nth power is n multiplied by x raised to the $(n-1)$th power.

Answer 1

Using the chain rule, $[(f(x))^n]'=n(f(x))^{n-1}f'(x)$. The chain rule is that $f(g(x))'=f'(g(x))g'(x)$, which is proved in any good calculus book