Calculus Examples

$y = \csc (2 x)$

Step 1

Find the first derivative.

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Step 1.1

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

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Step 1.1.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 1.1.2

The derivative of $\csc (u)$ with respect to $u$ is $- \csc (u) \cot (u)$ .

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

Step 1.1.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 1.2.2

Multiply $2$ by $- 1$ .

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

Step 1.2.3

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

$- 2 \csc (2 x) \cot (2 x) \cdot 1$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Multiply $- 2$ by $1$ .

$- 2 \csc (2 x) \cot (2 x)$

Step 1.2.4.2

Reorder the factors of $- 2 \csc (2 x) \cot (2 x)$ .

$f' (x) = - 2 \cot (2 x) \csc (2 x)$

Step 2

Find the second derivative.

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Step 2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 \cot (2 x) \csc (2 x)$ with respect to $x$ is $- 2 \frac{d}{d x} [\cot (2 x) \csc (2 x)]$ .

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

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \cot (2 x)$ and $g (x) = \csc (2 x)$ .

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

Step 2.3

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

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$- 2 (\cot (2 x) (\frac{d}{d u_{1}} [\csc (u_{1})] \frac{d}{d x} [2 x]) + \csc (2 x) \frac{d}{d x} [\cot (2 x)])$

Step 2.3.2

The derivative of $\csc (u_{1})$ with respect to $u_{1}$ is $- \csc (u_{1}) \cot (u_{1})$ .

$- 2 (\cot (2 x) (- \csc (u_{1}) \cot (u_{1}) \frac{d}{d x} [2 x]) + \csc (2 x) \frac{d}{d x} [\cot (2 x)])$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $2 x$ .

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

Step 2.4

Raise $\cot (2 x)$ to the power of $1$ .

$- 2 (- \csc (2 x) (\cot^{1} (2 x) \cot (2 x)) \frac{d}{d x} [2 x] + \csc (2 x) \frac{d}{d x} [\cot (2 x)])$

Step 2.5

Raise $\cot (2 x)$ to the power of $1$ .

$- 2 (- \csc (2 x) (\cot^{1} (2 x) \cot^{1} (2 x)) \frac{d}{d x} [2 x] + \csc (2 x) \frac{d}{d x} [\cot (2 x)])$

Step 2.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 2 (- \csc (2 x) {\cot (2 x)}^{1 + 1} \frac{d}{d x} [2 x] + \csc (2 x) \frac{d}{d x} [\cot (2 x)])$

Step 2.7

Differentiate.

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Step 2.7.1

Add $1$ and $1$ .

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

Step 2.7.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.7.3

Multiply $2$ by $- 1$ .

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

Step 2.7.4

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

$- 2 (- 2 \csc (2 x) \cot^{2} (2 x) \cdot 1 + \csc (2 x) \frac{d}{d x} [\cot (2 x)])$

Step 2.7.5

Multiply $- 2$ by $1$ .

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

Step 2.8

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 $g (x) = 2 x$ .

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Step 2.8.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

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

Step 2.8.2

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

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

Step 2.8.3

Replace all occurrences of $u_{2}$ with $2 x$ .

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

Step 2.9

Raise $\csc (2 x)$ to the power of $1$ .

$- 2 (- 2 \csc (2 x) \cot^{2} (2 x) - (\csc^{1} (2 x) \csc^{2} (2 x)) \frac{d}{d x} [2 x])$

Step 2.10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 2 (- 2 \csc (2 x) \cot^{2} (2 x) - {\csc (2 x)}^{1 + 2} \frac{d}{d x} [2 x])$

Step 2.11

Add $1$ and $2$ .

$- 2 (- 2 \csc (2 x) \cot^{2} (2 x) - \csc^{3} (2 x) \frac{d}{d x} [2 x])$

Step 2.12

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- 2 (- 2 \csc (2 x) \cot^{2} (2 x) - \csc^{3} (2 x) (2 \frac{d}{d x} [x]))$

Step 2.13

Multiply $2$ by $- 1$ .

$- 2 (- 2 \csc (2 x) \cot^{2} (2 x) - 2 \csc^{3} (2 x) \frac{d}{d x} [x])$

Step 2.14

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

$- 2 (- 2 \csc (2 x) \cot^{2} (2 x) - 2 \csc^{3} (2 x) \cdot 1)$

Step 2.15

Multiply $- 2$ by $1$ .

$- 2 (- 2 \csc (2 x) \cot^{2} (2 x) - 2 \csc^{3} (2 x))$

Step 2.16

Simplify.

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Step 2.16.1

Apply the distributive property.

$- 2 (- 2 \csc (2 x) \cot^{2} (2 x)) - 2 (- 2 \csc^{3} (2 x))$

Step 2.16.2

Combine terms.

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Step 2.16.2.1

Multiply $- 2$ by $- 2$ .

$4 (\csc (2 x) \cot^{2} (2 x)) - 2 (- 2 \csc^{3} (2 x))$

Step 2.16.2.2

Multiply $- 2$ by $- 2$ .

$4 \csc (2 x) \cot^{2} (2 x) + 4 \csc^{3} (2 x)$

Step 2.16.3

Reorder terms.

$f'' (x) = 4 \cot^{2} (2 x) \csc (2 x) + 4 \csc^{3} (2 x)$