Question

The derivative of $a^{Secx}$w.r.t. $a^{Tanx}(a>0)$ is

• A. $a^{Secx-Tanx}$
• B. $Sinx{a}^{Secx-Tanx}$
• C. $Sinx{a}^{Tanx-Secx}$
• D. $Secx{a}^{Secx-Tanx}$

Answer 1

Answer: (B)

Let $y=a^{\sec x}\text{ and }z=a^{\tan x}$
$\Rightarrow \frac{dy}{dx}=a^{\sec x}(\log a)(\sec x\tan x)$
and $\frac{dz}{dx}=a^{\tan x}(\log a)\left({\sec }^{2}x\right)$
$\therefore \frac{dy}{dz}=\frac{\frac{dy}{dx}}{\frac{dz}{dx}}=\frac{a^{\sec x}(\log a)(\sec x\tan x)}{a^{\tan x}(\log a)\left({\sec }^{2}x\right)}$
$=(\sin x)a^{\sec x-\tan x}$