Find the Derivative – d/dx cot(sin(x))^2

$\cot^{2} (\sin (x))$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cot (\sin (x))$ .

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To apply the Chain Rule, set $u_{1}$ as $\cot (\sin (x))$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\cot (\sin (x))]$

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1} \frac{d}{d x} [\cot (\sin (x))]$

Replace all occurrences of $u_{1}$ with $\cot (\sin (x))$ .

$2 \cot (\sin (x)) \frac{d}{d x} [\cot (\sin (x))]$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cot (x)$ and $g (x) = \sin (x)$ .

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To apply the Chain Rule, set $u_{2}$ as $\sin (x)$ .

$2 \cot (\sin (x)) (\frac{d}{d u_{2}} [\cot (u_{2})] \frac{d}{d x} [\sin (x)])$

The derivative of $\cot (u_{2})$ with respect to $u_{2}$ is $- \csc^{2} (u_{2})$ .

$2 \cot (\sin (x)) (- \csc^{2} (u_{2}) \frac{d}{d x} [\sin (x)])$

Replace all occurrences of $u_{2}$ with $\sin (x)$ .

$2 \cot (\sin (x)) (- \csc^{2} (\sin (x)) \frac{d}{d x} [\sin (x)])$

Multiply $- 1$ by $2$ .

$- 2 \cot (\sin (x)) (\csc^{2} (\sin (x)) \frac{d}{d x} [\sin (x)])$

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$- 2 \cot (\sin (x)) \csc^{2} (\sin (x)) \cos (x)$

Reorder the factors of $- 2 \cot (\sin (x)) \csc^{2} (\sin (x)) \cos (x)$ .

$- 2 \csc^{2} (\sin (x)) \cos (x) \cot (\sin (x))$

Find the Derivative – d/dx cot(sin(x))^2

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