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 $