Question #9eb92

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1 Answer
Bill K.
Sep 27, 2017

Use the Power Rule and Chain Rule to write #h'(x)=4/5 (f(x))^{-1/5} * f'(x)=(4f'(x))/(5 root(5){f(x)})#.

Explanation:

The Power Rule says that #d/(dx) (x^{n})=nx^{n-1}# for any number #n#. The Chain Rule says that #d/(dx)(g(f(x)))=g'(f(x)) * f'(x)# when #f# is differentiable at the value #x# and #g# is differentiable at the value #f(x)#.

For the problem at hand, #g(x)=root(5){x^{4}}=x^{4/5}# and therefore #g'(x)=4/5 x^{-1/5}=4/(5 root(5){x})#.

Therefore #h'(x)=d/(dx)(g(f(x)))=4/5 (f(x))^{-1/5} * f'(x)=(4f'(x))/(5 root(5){f(x)})#

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