Question

If $ y = log_{cos \,x} sin\, x $ , then $ \frac{dy}{dx} $ is equal to

• A. $ \frac{\left(cot \,x\, log \,cos\, x +tan \,x \,log\, sin\, x\right)}{\left(log \,cos\, x\right)^{2}} $
• B. $ \frac{\left(tan \,x\, log \,cos\, x +cot \,x \,log\, sin\, x\right)}{\left(log \,cos\, x\right)^{2}} $
• C. $ \frac{\left(cot \,x\, log \,cos\, x +tan \,x \,log\, sin\, x\right)}{\left(log \,sin\, x\right)^{2}} $
• D. None of the above

Answer 1

Answer: (A)

Given, $y=\log _{\cos x} \sin x=\frac{\log \sin x}{\log \cos x}$
On differentiating w.r.t. x, we get
$\frac{d y}{d x}=\frac{\cot x \cdot \log \cos x+\tan x \cdot \log \sin x}{(\log \cos x)^{2}}$