# How do you differentiate f(x)=csc^3(2x)?

Dec 14, 2016

$- 6 \tan 2 x {\csc}^{3} 2 x$

#### Explanation:

$f \left(x\right) = {\csc}^{3} \left(2 x\right) = {\left(\sin 2 x\right)}^{-} 3$

Now that we see the $\sin 2 x$ instead of $\csc 2 x$ term, it is pretty straight forward from here.

$\frac{\mathrm{df} \left(x\right)}{\mathrm{dx}} = - 3 {\left(\sin 2 x\right)}^{-} 4 \left(\cos 2 x\right) \left(2\right)$
$= - 6 \cot 2 x {\csc}^{3} 2 x$

-The $- 3$ comes from the power.
-The power $- 4$ is because of the differentiation.
-The $\cos 2 x$ because you want to differentiate $\sin 2 x$ in the bracket.
-The $2$ because you differentiate $2 x$.

Cheers