Question

Derivative of $log \left(sec\,\theta +tan \,\theta\right) $ with respect to $sec\, \theta$ at $\theta = \pi/4$ is

• A. $0$
• B. $1$
• C. $\frac{1}{\sqrt2}$
• D. $\sqrt2$

Answer 1

The Correct Option is B

Let $u=log (sec\, \theta+tan \, \theta)$ and $v=sec\, \theta$
On differentiating both sides w.r.t. $\theta,$ we get
$\frac{du}{d \theta}=\frac{1}{(\sec \theta+\tan \theta)}\left(\sec \theta \tan \theta+\sec ^{2} \theta\right)$
and $\frac{dv}{d \theta}=\sec \theta \tan \theta$
$\therefore \frac{du}{dv}=\frac{\frac{du}{d \theta}}{\frac{dv}{d \theta}}$
$=\frac{\sec \theta(\tan \theta+\sec \theta)}{(\sec \theta+\tan \theta) \times \sec \theta \tan \theta}=\cot \theta$
$\Rightarrow \frac{du}{dv\left(\theta=\frac{\pi}{4}\right)}=\cot \frac{\pi}{4}=1$