Question

The derivative of $a^{\sec x}$ w.r.t. $a^{\tan x}(a>0)$ is

• A. $\sec x{a}^{\sec x-\tan x}$
• B. $\sin x{a}^{\tan x-\sec x}$
• C. $\sin x{a}^{\sec x-\tan x}$
• D. $a^{\sec x-\tan x}$

Answer 1

Answer: (C)

Let $u=a^{\sec x}$ and $v=a^{\tan x}$ On differentiating w.r.t. $x,$ we get $\frac{du}{dx}=a^{\sec x}{\log }_{e}a.\sec x\tan x$ and $\frac{dv}{dx}=a^{\tan x}{\log }_{e}a.{\sec }^{2}x$ $\therefore$ $\frac{du}{dv}=\frac{a^{\sec x}{\log }_{e}a.\sec x\tan x}{a^{\tan x}{\log }_{e}a.{\sec }^{2}x}$ $=a^{\sec x-\tan x}\sin x$