need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.2 question 54

Answer:  $e^{a x} \sec x\left\{\mathrm{a} \tan 2 x+\tan x \tan 2 x+2 \sec ^{2} 2 x\right\}$

Hint:  you must know the rule of solving derivative of exponential and trigonometric functions

Given: $e^{a x} \sec x \tan 2 x$

Solution:

Let  $y=e^{a x} \sec x \tan 2 x$

Differentiate with respect to x

\begin{aligned} &\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}} e^{a x} \sec x \tan 2 x \\\\ &\frac{\mathrm{dy}}{\mathrm{dx}}=e^{a x} \frac{d}{d x}\{\sec x \tan 2 x\}+\sec x \tan 2 x \frac{d}{d x}\left\{e^{a x}\right\} \end{aligned}

\begin{aligned} &\frac{\mathrm{dy}}{\mathrm{dx}}=e^{a x}\left[\sec x \tan x \tan 2 x+2 \sec ^{2} 2 x \sec x\right]+a e^{a x} \sec x \tan 2 x \\\\ &\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{a} e^{a x} \sec x \tan 2 x+e^{a x} \sec x \tan x \tan 2 x+2 \sec ^{2} 2 x \sec x e^{a x} \end{aligned}

$\frac{\mathrm{dy}}{\mathrm{dx}}=e^{a x} \sec x\left\{\mathrm{a} \tan 2 x+\tan x \tan 2 x+2 \sec ^{2} 2 x\right\}$