Question
Diffrentiate the following w.r.t.x
$\log \left[\frac{a^{\cos x}}{\left(x^2-3\right)^3 \log x}\right]$
$\log \left[\frac{a^{\cos x}}{\left(x^2-3\right)^3 \log x}\right]$
Differentiating w.r.t. $x$, we get
$\frac{d y}{d x}=\frac{d}{d x}\left[(\cos x)(\log a)-3 \log \left(x^2-3\right)-\log (\log x)\right]$
$\begin{aligned} & =(\log a) \cdot \frac{d}{d x}(\cos x)-3 \frac{d}{d x}\left[\log \left(x^2-3\right)\right]-\frac{d}{d x}[\log (\log x)] \\ & =(\log a)(-\sin x)-3 \times \frac{1}{x^2-3} \cdot \frac{d}{d x}\left(x^2-3\right)-\frac{1}{\log x} \cdot \frac{d}{d x}(\log x) \\ & =-(\sin x)(\log a)-\frac{3}{x^2-3} \times(2 x-0)-\frac{1}{\log x} \times \frac{1}{x} \\ & =-(\sin x)(\log a)-\frac{6 x}{x^2-3}-\frac{1}{x \log x} .\end{aligned}$
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