Question

The derivative of $a^{Sec\,x}$w.r.t. $a^{Tan\,x}(a>0)$ is

• A. $a^{Secx-Tanx}$
• B. $Sin\,x\,a^{Secx-Tanx}$
• C. $Sin\,x\,a^{Tanx-Secx}$
• D. $Sec\,x a^{Secx-Tanx}$

Answer 1

Answer: (B)

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