Question

The differential coefficient of $\log \tan x$ is

Answer 1

Determine the differential coefficient.

Derivative of the function itself is called the differential coefficient of the function

Differentiate the function with respect to $x$:

$\begin{array}{rcl}\frac{d}{dx}\left(\log \left(\tan \left(x\right)\right)\right)& =& \frac{1}{\tan \left(x\right)}\times \frac{d}{dx}\left(\tan \left(x\right)\right)\\ & & \\ & \Rightarrow & \frac{d}{dx}\left(\log \left(\tan \left(x\right)\right)\right)=\frac{1}{\tan \left(x\right)}{\sec }^2\left(x\right)\\ & \Rightarrow & \frac{d}{dx}\left(\log \left(\tan \left(x\right)\right)\right)=\frac{\cos \left(x\right)}{\sin \left(x\right)}\frac{1}{{\cos }^2\left(x\right)}\\ & \Rightarrow & \frac{d}{dx}\left(\log \left(\tan \left(x\right)\right)\right)=\frac{1}{\sin \left(x\right)\cos \left(x\right)}\\ & \Rightarrow & \frac{d}{dx}\left(\log \left(\tan \left(x\right)\right)\right)=\frac{2}{\sin \left(2x\right)}\\ & & \end{array}$

So, the differential coefficient of $\log \tan x$ is $\frac{2}{\sin \left(2x\right)}\mathrm{or}2\csc \left(2x\right)$.