Question
Diffrentiate the following w.r.t.x
$\cos ^2\left[\log \left(x^2+7\right)\right]$
$\cos ^2\left[\log \left(x^2+7\right)\right]$
Differentiating w.r.t. x, we get
$\begin{aligned} & \frac{d y}{d x}=\frac{d}{d x}\left\{\cos \left[\log \left(x^2+7\right)\right]\right\}^2 \\ & =2 \cos \left[\log \left(x^2+7\right)\right] \cdot \frac{d}{d x}\left\{\cos \left[\log \left(x^2+7\right)\right]\right\} \\ & =2 \cos \left[\log \left(x^2+7\right)\right] \cdot\left\{-\sin \left[\log \left(x^2+7\right)\right]\right\} \cdot \frac{d}{d x}\left[\log \left(x^2+7\right)\right] \\ & =-2 \sin \left[\log \left(x^2+7\right)\right] \cdot \cos \left[\log \left(x^2+7\right)\right] \times \frac{1}{x^2+7} \cdot \frac{d}{d x}\left(x^2+7\right) \\ & =-\sin \left[2 \log \left(x^2+7\right)\right] \times \frac{1}{x^2+7} \cdot(2 x+0) \\ & =\frac{-2 x \cdot \sin \left[2 \log \left(x^2+7\right)\right]}{x^2+7} .\end{aligned}$
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