Question

Differentiate $\tan x^n+{\tan }^{n}x-{\tan }^{-1}\frac{a+x^n}{1-a{x}^n}.$

• A. $\left({\sec }^2{x}^n\right).n{x}^{n-1}+n{\tan }^{n-1}x.{\sec }^{2}x-\left[\frac{1}{(1-x^{2n})}\right]n{x}^{n-1}$
• B. $\left({\sec }^2{x}^n\right).n{x}^{n-1}+n{\tan }^{n-1}x.{\sec }^{2}x-\left[\frac{1}{(1+x^{2n})}\right]n{x}^n$
• C. $\left({\sec }^2{x}^n\right).n{x}^{n-1}+n{\tan }^{n}x.{\sec }^{2}x-\left[\frac{1}{(1+x^{2n})}\right]n{x}^{n-1}$
• D. $\left({\sec }^2{x}^n\right).n{x}^{n-1}+n{\tan }^{n-1}x.{\sec }^{2}x-\left[\frac{1}{(1+x^{2n})}\right]n{x}^{n-1}$

Answer 1

The correct option is D $\left({\sec }^2{x}^n\right).n{x}^{n-1}+n{\tan }^{n-1}x.{\sec }^{2}x-\left[\frac{1}{(1+x^{2n})}\right]n{x}^{n-1}$

Let $y=\tan x^n+{\tan }^{n}x-{\tan }^{-1}\frac{a+x^n}{1-a{x}^n}.$

$\Rightarrow y=\tan x^n+{\tan }^{n}x-({\tan }^{-1}a+{\tan }^{-1}{x}^n)$

$\therefore \frac{dy}{dx}=\left({\sec }^2{x}^n\right).n{x}^{n-1}+n{\tan }^{n-1}x.{\sec }^{2}x-0-\left[\frac{1}{(1+x^{2n})}\right]n{x}^{n-1}$

$=\left({\sec }^2{x}^n\right).n{x}^{n-1}+n{\tan }^{n-1}x.{\sec }^{2}x-\left[\frac{1}{(1+x^{2n})}\right]n{x}^{n-1}$