Question

If $y={\log }_{\sin x}\left(\tan x\right)$, then ${\left(\frac{dy}{dx}\right)}_{x=\frac{\pi }{4}}$ is

• A. $\frac{4}{\ln 2}$
• B. $-4\ln 2$
• C. $-\frac{4}{\ln 2}$
• D. $4\ln 2$

Answer 1

The correct option is C $-\frac{4}{\ln 2}$

$y=\frac{\ln \left(\tan x\right)}{\ln \left(\sin x\right)}$

$\frac{dy}{dx}=\frac{\left(\ln \sin x\right)\left(\frac{{\sec }^{2}x}{\tan x}\right)-\ln \left(\tan x\right).\cot x}{(\ln \sin x{)}^2}$

$\frac{dy}{dx}{|}_{x=\frac{\pi }{4}}=-\frac{4}{\ln 2}$