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

Dec 21, 2016

$f ' \left(g \left(x\right)\right) = \textcolor{g r e e n}{5}$
or
f'(g(x))=color(green)(10/root(3)(x)

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

#### Explanation:

Version 1
If $f \left(x\right) = 5 x$
then $f ' \left(x\right) = 5$ (Exponent rule for derivatives)

That is $f ' \left(x\right)$ is a constant ($5$) for any value of $x$.

Specifically if we replace $x$ with $g \left(x\right)$
$f ' \left(g \left(x\right)\right)$ is still equal to the constant $5$

Version 2
If $f \left(x\right) = 5 x$ and $g \left(x\right) = 3 {x}^{\frac{2}{3}}$
then $f \left(g \left(x\right)\right) = 15 {x}^{2} / 3$
and
$f ' \left(g \left(x\right)\right) = \frac{2}{3} \times 5 {x}^{- \frac{1}{3}} = \frac{10}{x} ^ \left(\frac{1}{3}\right) = \frac{10}{\sqrt[3]{x}}$