# How do I find d/dxln (tan (x^2))?

Jan 28, 2015

I would use the Chain Rule.
In a way it is like those russian dolls called Matrioskas where you have a smaller doll inside another. To get to the last one you have to open the big ones first.

(Picture source: https://ghaiaesoterico.wordpress.com/2013/08/20/lenda-de-matrioska/)

Here the smallest doll is the function ${x}^{2}$ but to get to it you have first to "open" $\tan$ and $\ln$.

To open in this case means to derive.

So you get:

$\frac{d}{\mathrm{dx}} \ln \left(\tan \left({x}^{2}\right)\right) = \frac{1}{\tan \left({x}^{2}\right)} \cdot \frac{1}{{\cos}^{2} \left({x}^{2}\right)} \cdot 2 x$

Remember that:
$\frac{d}{\mathrm{dx}} \ln \left(x\right) = \frac{1}{x}$
$\frac{d}{\mathrm{dx}} \tan \left(x\right) = \frac{1}{{\cos}^{2} \left(x\right)}$
$\frac{d}{\mathrm{dx}} {x}^{2} = 2 x$