Q80 / 150 · MHT CET 2007 · Mathematics

If $y=\log _{\cos x} \sin x$, then $\frac{d y}{d x}$ is equal to

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

Solution

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

Hence, the correct answer is option: 'A'.

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