Differentiate the following w.r.t.x: ,x>0 - Vidyadip

Question

Differentiate the following w.r.t.x: $\frac{\cos\text{x}}{\log\text{x}},\text{x}>0$

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Answer

$\text{Let y}=\frac{\cos\text{x}}{\log\text{x}}$
$\therefore\ \frac{\text{dy}}{\text{dx}} =\frac{(\log\text{x}).\frac{\text{d}}{\text{dx}}(\cos\text{x})-\cos\text{x}.\frac{\text{d}}{\text{dx}}(\log\text{x})}{(\log\text{x})^{2}}$
$ =\frac{(\log\text{x}).(-\sin\text{x})-\cos\text{x}\cdot\frac{\text{1}}{\text{x}}}{(\log\text{x})^{2}}=\frac{-\sin\text{x}\log\text{x}-\frac{\cos\text{x}}{\text{x}}}{(\log\text{x}^{2})}$
$=\frac{-\text{x}\sin\text{x}\ \log\text{x}-\cos\text{x}}{\text{x}(\log\text{x})^2}$

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