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

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Answer 1 · 2016-12-21T11:30:36

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

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

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
$f'(g(x)) = 2/3xx5x^(-1/3)=10/x^(1/3)=10/root(3)(x)$

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