Question

If $y=\log \sqrt{\tan x}$, then the value of $\frac{dy}{dx}$ at $x=\frac{\pi }{4}$ is given by

• A. $1$
• B. $0$
• C. $\frac{1}{2}$
• D. $\infty$

Answer 1

Answer: (A)

$1$

$y=\log \sqrt{\tan x}$

$dfracdydx=\frac{d}{dx}\log \sqrt{\tan x}$

$=\frac{d\log \sqrt{\tan x}}{d\sqrt{\tan x}}\times \frac{d\sqrt{\tan x}}{d\tan x}\times \frac{d\tan x}{dx}$

Taking $z=\sqrt{\tan x}$ and $t=\tan x$

$\frac{dy}{dx}=\frac{d\log z}{dz}\times \frac{d\sqrt{t}}{dt}\times {\sec }^{2}x$

$=\frac{1}{z}\times \frac{1}{2\sqrt{t}}\times {\sec }^{2}x$

$=\frac{{\sec }^{2}x}{2\sqrt{\tan x}\times \sqrt{\tan x}}=\frac{{\sec }^{2}x}{2\tan x}$

At $x=\frac{\pi }{4}$

$\frac{dy}{dx}=\frac{{\sec }^2\frac{\pi }{4}}{2\tan \frac{\pi }{4}}$

$=\frac{(\sqrt{2}{)}^2}{2\times 1}=\frac{2}{2}$

$⟹\frac{dy}{dx}=1$

Answer-(A)