Calculus Examples

$y = \frac{1}{36} \cot (6 x + 5)$

Step 1

Find the first derivative.

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Step 1.1

Since $\frac{1}{36}$ is constant with respect to $x$ , the derivative of $\frac{1}{36} \cdot \cot (6 x + 5)$ with respect to $x$ is $\frac{1}{36} \frac{d}{d x} [\cot (6 x + 5)]$ .

$\frac{1}{36} \frac{d}{d x} [\cot (6 x + 5)]$

Step 1.2

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 $g (x) = 6 x + 5$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $6 x + 5$ .

$\frac{1}{36} (\frac{d}{d u} [\cot (u)] \frac{d}{d x} [6 x + 5])$

Step 1.2.2

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

$\frac{1}{36} (- \csc^{2} (u) \frac{d}{d x} [6 x + 5])$

Step 1.2.3

Replace all occurrences of $u$ with $6 x + 5$ .

$\frac{1}{36} (- \csc^{2} (6 x + 5) \frac{d}{d x} [6 x + 5])$

Step 1.3

Differentiate.

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Step 1.3.1

Combine $\frac{1}{36}$ and $\csc^{2} (6 x + 5)$ .

$- \frac{\csc^{2} (6 x + 5)}{36} \frac{d}{d x} [6 x + 5]$

Step 1.3.2

By the Sum Rule, the derivative of $6 x + 5$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [5]$ .

$- \frac{\csc^{2} (6 x + 5)}{36} (\frac{d}{d x} [6 x] + \frac{d}{d x} [5])$

Step 1.3.3

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$- \frac{\csc^{2} (6 x + 5)}{36} (6 \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 1.3.4

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

$- \frac{\csc^{2} (6 x + 5)}{36} (6 \cdot 1 + \frac{d}{d x} [5])$

Step 1.3.5

Multiply $6$ by $1$ .

$- \frac{\csc^{2} (6 x + 5)}{36} (6 + \frac{d}{d x} [5])$

Step 1.3.6

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$- \frac{\csc^{2} (6 x + 5)}{36} (6 + 0)$

Step 1.3.7

Simplify terms.

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Step 1.3.7.1

Add $6$ and $0$ .

$- \frac{\csc^{2} (6 x + 5)}{36} \cdot 6$

Step 1.3.7.2

Multiply $6$ by $- 1$ .

$- 6 \frac{\csc^{2} (6 x + 5)}{36}$

Step 1.3.7.3

Combine $- 6$ and $\frac{\csc^{2} (6 x + 5)}{36}$ .

$\frac{- 6 \csc^{2} (6 x + 5)}{36}$

Step 1.3.7.4

Cancel the common factor of $- 6$ and $36$ .

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Step 1.3.7.4.1

Factor $6$ out of $- 6 \csc^{2} (6 x + 5)$ .

$\frac{6 (- \csc^{2} (6 x + 5))}{36}$

Step 1.3.7.4.2

Cancel the common factors.

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Step 1.3.7.4.2.1

Factor $6$ out of $36$ .

$\frac{6 (- \csc^{2} (6 x + 5))}{6 (6)}$

Step 1.3.7.4.2.2

Cancel the common factor.

$\frac{6 (- \csc^{2} (6 x + 5))}{6 \cdot 6}$

Step 1.3.7.4.2.3

Rewrite the expression.

$\frac{- \csc^{2} (6 x + 5)}{6}$

Step 1.3.7.5

Move the negative in front of the fraction.

$f' (x) = - \frac{\csc^{2} (6 x + 5)}{6}$

Step 2

Find the second derivative.

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Step 2.1

Since $- \frac{1}{6}$ is constant with respect to $x$ , the derivative of $- \frac{\csc^{2} (6 x + 5)}{6}$ with respect to $x$ is $- \frac{1}{6} \frac{d}{d x} [\csc^{2} (6 x + 5)]$ .

$- \frac{1}{6} \frac{d}{d x} [\csc^{2} (6 x + 5)]$

Step 2.2

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 $g (x) = \csc (6 x + 5)$ .

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $\csc (6 x + 5)$ .

$- \frac{1}{6} (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\csc (6 x + 5)])$

Step 2.2.2

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

$- \frac{1}{6} (2 u_{1} \frac{d}{d x} [\csc (6 x + 5)])$

Step 2.2.3

Replace all occurrences of $u_{1}$ with $\csc (6 x + 5)$ .

$- \frac{1}{6} (2 \csc (6 x + 5) \frac{d}{d x} [\csc (6 x + 5)])$

Step 2.3

Simplify terms.

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Step 2.3.1

Multiply $2$ by $- 1$ .

$- 2 (\frac{1}{6}) (\csc (6 x + 5) \frac{d}{d x} [\csc (6 x + 5)])$

Step 2.3.2

Combine $- 2$ and $\frac{1}{6}$ .

$\frac{- 2}{6} (\csc (6 x + 5) \frac{d}{d x} [\csc (6 x + 5)])$

Step 2.3.3

Combine $\csc (6 x + 5)$ and $\frac{- 2}{6}$ .

$\frac{\csc (6 x + 5) \cdot - 2}{6} \frac{d}{d x} [\csc (6 x + 5)]$

Step 2.3.4

Move $- 2$ to the left of $\csc (6 x + 5)$ .

$\frac{- 2 \cdot \csc (6 x + 5)}{6} \frac{d}{d x} [\csc (6 x + 5)]$

Step 2.3.5

Cancel the common factor of $- 2$ and $6$ .

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Step 2.3.5.1

Factor $2$ out of $- 2 \csc (6 x + 5)$ .

$\frac{2 (- \csc (6 x + 5))}{6} \frac{d}{d x} [\csc (6 x + 5)]$

Step 2.3.5.2

Cancel the common factors.

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Step 2.3.5.2.1

Factor $2$ out of $6$ .

$\frac{2 (- \csc (6 x + 5))}{2 (3)} \frac{d}{d x} [\csc (6 x + 5)]$

Step 2.3.5.2.2

Cancel the common factor.

$\frac{2 (- \csc (6 x + 5))}{2 \cdot 3} \frac{d}{d x} [\csc (6 x + 5)]$

Step 2.3.5.2.3

Rewrite the expression.

$\frac{- \csc (6 x + 5)}{3} \frac{d}{d x} [\csc (6 x + 5)]$

Step 2.3.6

Move the negative in front of the fraction.

$- \frac{\csc (6 x + 5)}{3} \frac{d}{d x} [\csc (6 x + 5)]$

Step 2.4

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 $g (x) = 6 x + 5$ .

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Step 2.4.1

To apply the Chain Rule, set $u_{2}$ as $6 x + 5$ .

$- \frac{\csc (6 x + 5)}{3} (\frac{d}{d u_{2}} [\csc (u_{2})] \frac{d}{d x} [6 x + 5])$

Step 2.4.2

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

$- \frac{\csc (6 x + 5)}{3} (- \csc (u_{2}) \cot (u_{2}) \frac{d}{d x} [6 x + 5])$

Step 2.4.3

Replace all occurrences of $u_{2}$ with $6 x + 5$ .

$- \frac{\csc (6 x + 5)}{3} (- \csc (6 x + 5) \cot (6 x + 5) \frac{d}{d x} [6 x + 5])$

Step 2.5

Combine fractions.

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Step 2.5.1

Multiply $- 1$ by $- 1$ .

$1 \frac{\csc (6 x + 5)}{3} (\csc (6 x + 5) \cot (6 x + 5) \frac{d}{d x} [6 x + 5])$

Step 2.5.2

Multiply $\frac{\csc (6 x + 5)}{3}$ by $1$ .

$\frac{\csc (6 x + 5)}{3} (\csc (6 x + 5) \cot (6 x + 5) \frac{d}{d x} [6 x + 5])$

Step 2.5.3

Combine $\csc (6 x + 5)$ and $\frac{\csc (6 x + 5)}{3}$ .

$\frac{\csc (6 x + 5) \csc (6 x + 5)}{3} (\cot (6 x + 5) \frac{d}{d x} [6 x + 5])$

Step 2.6

Raise $\csc (6 x + 5)$ to the power of $1$ .

$\frac{\csc^{1} (6 x + 5) \csc (6 x + 5)}{3} (\cot (6 x + 5) \frac{d}{d x} [6 x + 5])$

Step 2.7

Raise $\csc (6 x + 5)$ to the power of $1$ .

$\frac{\csc^{1} (6 x + 5) \csc^{1} (6 x + 5)}{3} (\cot (6 x + 5) \frac{d}{d x} [6 x + 5])$

Step 2.8

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

$\frac{{\csc (6 x + 5)}^{1 + 1}}{3} (\cot (6 x + 5) \frac{d}{d x} [6 x + 5])$

Step 2.9

Combine fractions.

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Step 2.9.1

Add $1$ and $1$ .

$\frac{\csc^{2} (6 x + 5)}{3} (\cot (6 x + 5) \frac{d}{d x} [6 x + 5])$

Step 2.9.2

Combine $\cot (6 x + 5)$ and $\frac{\csc^{2} (6 x + 5)}{3}$ .

$\frac{\cot (6 x + 5) \csc^{2} (6 x + 5)}{3} \frac{d}{d x} [6 x + 5]$

Step 2.10

By the Sum Rule, the derivative of $6 x + 5$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [5]$ .

$\frac{\cot (6 x + 5) \csc^{2} (6 x + 5)}{3} (\frac{d}{d x} [6 x] + \frac{d}{d x} [5])$

Step 2.11

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$\frac{\cot (6 x + 5) \csc^{2} (6 x + 5)}{3} (6 \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 2.12

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

$\frac{\cot (6 x + 5) \csc^{2} (6 x + 5)}{3} (6 \cdot 1 + \frac{d}{d x} [5])$

Step 2.13

Multiply $6$ by $1$ .

$\frac{\cot (6 x + 5) \csc^{2} (6 x + 5)}{3} (6 + \frac{d}{d x} [5])$

Step 2.14

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{\cot (6 x + 5) \csc^{2} (6 x + 5)}{3} (6 + 0)$

Step 2.15

Simplify terms.

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Step 2.15.1

Add $6$ and $0$ .

$\frac{\cot (6 x + 5) \csc^{2} (6 x + 5)}{3} \cdot 6$

Step 2.15.2

Combine $\frac{\cot (6 x + 5) \csc^{2} (6 x + 5)}{3}$ and $6$ .

$\frac{\cot (6 x + 5) \csc^{2} (6 x + 5) \cdot 6}{3}$

Step 2.15.3

Move $6$ to the left of $\cot (6 x + 5) \csc^{2} (6 x + 5)$ .

$\frac{6 \cdot (\cot (6 x + 5) \csc^{2} (6 x + 5))}{3}$

Step 2.15.4

Cancel the common factor of $6$ and $3$ .

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Step 2.15.4.1

Factor $3$ out of $6 \cot (6 x + 5) \csc^{2} (6 x + 5)$ .

$\frac{3 (2 \cot (6 x + 5) \csc^{2} (6 x + 5))}{3}$

Step 2.15.4.2

Cancel the common factors.

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Step 2.15.4.2.1

Factor $3$ out of $3$ .

$\frac{3 (2 \cot (6 x + 5) \csc^{2} (6 x + 5))}{3 (1)}$

Step 2.15.4.2.2

Cancel the common factor.

$\frac{3 (2 \cot (6 x + 5) \csc^{2} (6 x + 5))}{3 \cdot 1}$

Step 2.15.4.2.3

Rewrite the expression.

$\frac{2 \cot (6 x + 5) \csc^{2} (6 x + 5)}{1}$

Step 2.15.4.2.4

Divide $2 \cot (6 x + 5) \csc^{2} (6 x + 5)$ by $1$ .

$2 \cot (6 x + 5) \csc^{2} (6 x + 5)$

Step 2.15.5

Reorder the factors of $2 \cot (6 x + 5) \csc^{2} (6 x + 5)$ .

$f'' (x) = 2 \csc^{2} (6 x + 5) \cot (6 x + 5)$