Calculus Examples

\cot^{2} (x)

$\cot^{2} (x)$

Step 1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = x^{2}

$f (x) = x^{2}$ and

g (x) = \cot (x)

$g (x) = \cot (x)$ .

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Step 1.1

To apply the Chain Rule, set

u

$u$ as

\cot (x)

$\cot (x)$ .

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

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

Step 1.2

Differentiate using the Power Rule which states that

\frac{d}{d u} [u^{n}]

$\frac{d}{d u} [u^{n}]$ is

n u^{n - 1}

$n u^{n - 1}$ where

n = 2

$n = 2$ .

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

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

Step 1.3

Replace all occurrences of

u

$u$ with

\cot (x)

$\cot (x)$ .

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

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

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

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

Step 2

The derivative of

\cot (x)

$\cot (x)$ with respect to

x

$x$ is

- \csc^{2} (x)

$- \csc^{2} (x)$ .

2 \cot (x) (- \csc^{2} (x))

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

Step 3

Simplify the expression.

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Step 3.1

Multiply

- 1

$- 1$ by

2

$2$ .

- 2 \cot (x) \csc^{2} (x)

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

Step 3.2

Reorder the factors of

- 2 \cot (x) \csc^{2} (x)

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

- 2 \csc^{2} (x) \cot (x)

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

- 2 \csc^{2} (x) \cot (x)

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

\cot^{2} x

$\cot^{2} x$

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