Calculus Examples

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

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $\cot (\sin (x))$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\cot (\sin (x))$ .

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

Step 2

Rewrite $\cot (\sin (x))$ in terms of sines and cosines.

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

Step 3

Convert from $\frac{\cos (\sin (x))}{\sin (\sin (x))}$ to $\cot (\sin (x))$ .

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

Step 4

Since $\cot (\sin (x))$ is constant with respect to $z$ , the derivative of $\cot (\sin (x))$ with respect to $z$ is $0$ .

$2 \cot (\sin (x)) \cdot 0$

Step 5

Combine terms.

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Step 5.1

Multiply $0$ by $2$ .

$0 \cot (\sin (x))$

Step 5.2

Multiply $0$ by $\cot (\sin (x))$ .

$0$