If #f(x)= 5x # and #g(x) = 3x^( 2/3 ) #, what is #f'(g(x)) #?

1 Answer
Dec 21, 2016

Answer:

#f'(g(x))=color(green)(5)#
or
#f'(g(x))=color(green)(10/root(3)(x)#

perhaps depending upon the interpretation of #f'(g(x))#

Explanation:

Version 1
If #f(x)=5x#
then #f'(x)=5# (Exponent rule for derivatives)

That is #f'(x)# is a constant (#5#) for any value of #x#.

Specifically if we replace #x# with #g(x)#
#f'(g(x))# is still equal to the constant #5#

Version 2
If #f(x)=5x# and #g(x)=3x^(2/3)#
then #f(g(x)) =15x^2/3#
and
#f'(g(x)) = 2/3xx5x^(-1/3)=10/x^(1/3)=10/root(3)(x)#