# How do you differentiate tan(3x^2) - csc ( ln(4x) )^2?

Jun 23, 2015

$6 x {\sec}^{2} \left(3 {x}^{2}\right) + \frac{2 \cdot {\csc}^{2} \left(\ln \left(4 x\right)\right) \cdot \cot \left(\ln \left(4 x\right)\right)}{x}$

#### Explanation:

Wow, this looks tough. Let's break it down piece by piece.

First, let's take a look at the $\tan \left(3 {x}^{2}\right)$ term. We apply the chain rule when we differentiate this term.

We first differentiate this term with respect to the inner function $3 {x}^{2}$ and then multiply the result by the derivative of $3 {x}^{2}$ with respect to $x$ (By the chain rule).

${\left(\tan \left(3 {x}^{2}\right)\right)}^{'} = {\sec}^{2} \left(3 {x}^{2}\right) \cdot \left(6 x\right) = 6 x {\sec}^{2} \left(3 {x}^{2}\right)$

(Remember: $\frac{d}{\mathrm{dx}} \left(\tan x\right) = {\sec}^{2} \left(x\right)$)

Now for the other (very scary) term, $- {\csc}^{2} \left(\ln \left(4 x\right)\right)$.

Wait a minute, that's not too bad. There are just four functions nested in each other; the $4 x$ in the $\ln \left(\right)$, the $\ln \left(\right)$ in the $\csc \left(\right)$ and finally the $\csc \left(\right)$ in the ${\left(\right)}^{2}$.

This means that we will need to apply the chain rule three times.

First we see that,

${\left(- {\csc}^{2} \left(\ln \left(4 x\right)\right)\right)}^{'} = \left[- 2 \csc \left(\ln \left(4 x\right)\right)\right] \cdot {\left[\csc \left(\ln \left(4 x\right)\right)\right]}^{'}$

Now,

${\left[\csc \left(\ln \left(4 x\right)\right)\right]}^{'} = \left[- \csc \left(\ln \left(4 x\right)\right) \cdot \cot \left(\ln \left(4 x\right)\right)\right] {\left[\ln \left(4 x\right)\right]}^{'}$

and

${\left[\ln \left(4 x\right)\right]}^{'} = \left[\frac{1}{4 x}\right] \cdot {\left[4 x\right]}^{'} = 4 \left(\frac{1}{4 x}\right) = \frac{1}{x}$

Putting everything together we have

-csc^2(ln(4x)))^'=([-2csc(ln(4x))]*[-csc(ln(4x))*cot(ln(4x))])/x
$= \frac{2 \cdot {\csc}^{2} \left(\ln \left(4 x\right)\right) \cdot \cot \left(\ln \left(4 x\right)\right)}{x}$

So finally, adding on $6 x {\sec}^{2} \left(3 {x}^{2}\right)$, we have the derivative of the given expression as

$6 x {\sec}^{2} \left(3 {x}^{2}\right) + \frac{2 \cdot {\csc}^{2} \left(\ln \left(4 x\right)\right) \cdot \cot \left(\ln \left(4 x\right)\right)}{x}$

Woohoo! :)