Differentiate the following w.r.t. x:
Question:

Differentiate the following w.r.t. x:

$\frac{\cos x}{\log x}, x>0$

Solution:

Let $y=\frac{\cos x}{\log x}$

By using the quotient rule, we obtainc

$\frac{d y}{d x}=\frac{\frac{d}{d x}(\cos x) \times \log x-\cos x \times \frac{d}{d x}(\log x)}{(\log x)^{2}}$

$=\frac{-\sin x \log x-\cos x \times \frac{1}{x}}{(\log x)^{2}}$

$=\frac{-[x \log x \cdot \sin x+\cos x]}{x(\log x)^{2}}, x>0$

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