How to differentiate fraction?

how do i differentiate this? first principles not required.

Apr 21, 2018

Chain rule can be used once the fraction is simplified.

Explanation:

Simplify:

$\frac{{\left(4 x + 1\right)}^{2}}{\sqrt{4 x + 1}}$
$= \frac{{\left(4 x + 1\right)}^{2}}{{\left(4 x + 1\right)}^{\frac{1}{2}}}$

Exponent Law ${a}^{n} / {a}^{m} = {a}^{n - m}$

$= {\left(4 x + 1\right)}^{2 - \frac{1}{2}}$
$= {\left(4 x + 1\right)}^{\frac{3}{2}}$

Now apply chain rule:

$= \left(\frac{3}{2}\right) \left(4\right) {\left(4 x + 1\right)}^{\frac{1}{2}}$
$= 6 \sqrt{4 x + 1}$