Diffrentiate the following w.r.t.x ^3 [ (x^3 ) ] - Vidyadip

Question

Diffrentiate the following w.r.t.x

$\cot ^3\left[\log \left(x^3\right)\right]$

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Answer

Let $y=\cot ^3\left[\log \left(x^3\right)\right]$

Differentiating w.r.t. x, we get

$\begin{aligned} \frac{d y}{d x} & =\frac{d}{d x}\left[\cot \left(\log x^3\right)\right]^3 \\ & =3\left[\cot \left(\log x^3\right)\right]^2 \cdot \frac{d}{d x}\left[\cot \left(\log x^3\right)\right] \\ & =3 \cot ^2\left[\log \left(x^3\right)\right] \cdot\left[-\operatorname{cosec}^2\left(\log x^3\right)\right] \cdot \frac{d}{d x}\left(\log x^3\right) \\ & =-3 \cot ^2\left[\log \left(x^3\right)\right] \cdot \operatorname{cosec} 2\left[\log \left(x^3\right)\right] \cdot 3 \frac{d}{d x}(\log x) \\ & =-3 \cot ^2\left[\log \left(x^3\right)\right] \cdot \operatorname{cosec}^2\left[\log \left(x^3\right)\right] \cdot 3 \times \frac{1}{x} \\ & =\frac{-9 \operatorname{cosec}^2\left[\log \left(x^3\right)\right] \cdot \cot ^2\left[\log \left(x^3\right)\right]}{x} .\end{aligned}$

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