Derivative of log (secθ + tanθ ) with respect to secθ at θ = π/4 is ____________

Question

Derivative of log (secθ + tanθ ) with respect to secθ at θ = \(π\over4\) is ____________ 

A) 0 B) 1 C) \(\frac{1}{\sqrt2}\) D) \(\sqrt2\)

Answer 1

The derivative of log(secθ+tanθ) with respect to secθ at θ= $$\frac{\pi}{4}$$ is:

$$\frac{d}{d\sec\theta}\left[\log(\sec\theta+\tan\theta)\right]_{\theta=\frac{\pi}{4}}=\frac{\tan\theta}{\sec\theta+\tan\theta}\bigg|_{\theta=\frac{\pi}{4}}=\frac{\tan\frac{\pi}{4}}{\sec\frac{\pi}{4}+\tan\frac{\pi}{4}}=\frac{1}{\sqrt{2}+1}=\sqrt{2}-1$$

Therefore, the answer is D) √2.