Question

Differentiate $\log \sqrt{\frac{1+{\cos }^{2}x}{(1-e^{2x})}}$ w.r.t. $x$.

Answer 1

Derivative of $\log \sqrt{\frac{1+{\cos }^{2}x}{(1-e^{2x})}}$ can be easily evaluated using chain rule

Derivative of $\log \sqrt{\frac{1+{\cos }^{2}x}{(1-e^{2x})}}$ =$\left(\frac{1-e^{2x}}{1+co{s}^{2}x}\right)\times$ Derivative of $\frac{1+co{s}^{2}x}{1-e^{2x}}$ wrt x

$\to$ $\left(\frac{1-e^{2x}}{1+co{s}^{2}x}\right)\times$ $\frac{(1-e^{2x})(-Sin2x)-(1+co{s}^{2}x)(-2e^{2x})}{(1-e^{2x}{)}^2}$

$\to$ $\frac{(1-e^{2x})(-Sin2x)-(1+co{s}^{2}x)(-2e^{2x})}{(1-e^{2x})(1+co{s}^{2}x)}$