Calculus Examples

$f (x) = \frac{\cot (2 x)}{\csc (5 x)}$

Step 1

Rewrite $\csc (5 x)$ in terms of sines and cosines.

$\frac{d}{d x} [\frac{\cot (2 x)}{\frac{1}{\sin (5 x)}}]$

Step 2

Rewrite $\cot (2 x)$ in terms of sines and cosines.

$\frac{d}{d x} [\frac{\frac{\cos (2 x)}{\sin (2 x)}}{\frac{1}{\sin (5 x)}}]$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (5 x)}$ .

$\frac{d}{d x} [\frac{\cos (2 x)}{\sin (2 x)} \sin (5 x)]$

Step 4

Write $\sin (5 x)$ as a fraction with denominator $1$ .

$\frac{d}{d x} [\frac{\cos (2 x)}{\sin (2 x)} \cdot \frac{\sin (5 x)}{1}]$

Step 5

Simplify.

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Step 5.1

Divide $\sin (5 x)$ by $1$ .

$\frac{d}{d x} [\frac{\cos (2 x)}{\sin (2 x)} \sin (5 x)]$

Step 5.2

Convert from $\frac{\cos (2 x)}{\sin (2 x)}$ to $\cot (2 x)$ .

$\frac{d}{d x} [\cot (2 x) \sin (5 x)]$

Step 6

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 $g (x) = \sin (5 x)$ .

$\cot (2 x) \frac{d}{d x} [\sin (5 x)] + \sin (5 x) \frac{d}{d x} [\cot (2 x)]$

Step 7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 5 x$ .

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Step 7.1

To apply the Chain Rule, set $u_{1}$ as $5 x$ .

$\cot (2 x) (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [5 x]) + \sin (5 x) \frac{d}{d x} [\cot (2 x)]$

Step 7.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cot (2 x) (\cos (u_{1}) \frac{d}{d x} [5 x]) + \sin (5 x) \frac{d}{d x} [\cot (2 x)]$

Step 7.3

Replace all occurrences of $u_{1}$ with $5 x$ .

$\cot (2 x) (\cos (5 x) \frac{d}{d x} [5 x]) + \sin (5 x) \frac{d}{d x} [\cot (2 x)]$

Step 8

Differentiate.

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Step 8.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\cot (2 x) \cos (5 x) (5 \frac{d}{d x} [x]) + \sin (5 x) \frac{d}{d x} [\cot (2 x)]$

Step 8.2

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

$\cot (2 x) \cos (5 x) (5 \cdot 1) + \sin (5 x) \frac{d}{d x} [\cot (2 x)]$

Step 8.3

Simplify the expression.

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Step 8.3.1

Multiply $5$ by $1$ .

$\cot (2 x) \cos (5 x) \cdot 5 + \sin (5 x) \frac{d}{d x} [\cot (2 x)]$

Step 8.3.2

Move $5$ to the left of $\cot (2 x) \cos (5 x)$ .

$5 \cdot (\cot (2 x) \cos (5 x)) + \sin (5 x) \frac{d}{d x} [\cot (2 x)]$

Step 9

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 9.1

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

$5 \cot (2 x) \cos (5 x) + \sin (5 x) (\frac{d}{d u_{2}} [\cot (u_{2})] \frac{d}{d x} [2 x])$

Step 9.2

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

$5 \cot (2 x) \cos (5 x) + \sin (5 x) (- \csc^{2} (u_{2}) \frac{d}{d x} [2 x])$

Step 9.3

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

$5 \cot (2 x) \cos (5 x) + \sin (5 x) (- \csc^{2} (2 x) \frac{d}{d x} [2 x])$

Step 10

Differentiate.

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Step 10.1

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

$5 \cot (2 x) \cos (5 x) + \sin (5 x) (- \csc^{2} (2 x) (2 \frac{d}{d x} [x]))$

Step 10.2

Multiply $2$ by $- 1$ .

$5 \cot (2 x) \cos (5 x) + \sin (5 x) (- 2 \csc^{2} (2 x) \frac{d}{d x} [x])$

Step 10.3

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

$5 \cot (2 x) \cos (5 x) + \sin (5 x) (- 2 \csc^{2} (2 x) \cdot 1)$

Step 10.4

Multiply $- 2$ by $1$ .

$5 \cot (2 x) \cos (5 x) + \sin (5 x) (- 2 \csc^{2} (2 x))$

Step 11

Simplify.

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Step 11.1

Reorder terms.

$5 \cos (5 x) \cot (2 x) - 2 \csc^{2} (2 x) \sin (5 x)$

Step 11.2

Simplify each term.

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Step 11.2.1

Rewrite $\cot (2 x)$ in terms of sines and cosines.

$5 \cos (5 x) \frac{\cos (2 x)}{\sin (2 x)} - 2 \csc^{2} (2 x) \sin (5 x)$

Step 11.2.2

Multiply $5 \cos (5 x) \frac{\cos (2 x)}{\sin (2 x)}$ .

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Step 11.2.2.1

Combine $\frac{\cos (2 x)}{\sin (2 x)}$ and $5$ .

$\frac{\cos (2 x) \cdot 5}{\sin (2 x)} \cos (5 x) - 2 \csc^{2} (2 x) \sin (5 x)$

Step 11.2.2.2

Combine $\frac{\cos (2 x) \cdot 5}{\sin (2 x)}$ and $\cos (5 x)$ .

$\frac{\cos (2 x) \cdot 5 \cos (5 x)}{\sin (2 x)} - 2 \csc^{2} (2 x) \sin (5 x)$

Step 11.2.3

Move $5$ to the left of $\cos (2 x)$ .

$\frac{5 \cdot \cos (2 x) \cos (5 x)}{\sin (2 x)} - 2 \csc^{2} (2 x) \sin (5 x)$

Step 11.2.4

Rewrite $\csc (2 x)$ in terms of sines and cosines.

$\frac{5 \cos (2 x) \cos (5 x)}{\sin (2 x)} - 2 {(\frac{1}{\sin (2 x)})}^{2} \sin (5 x)$

Step 11.2.5

Apply the product rule to $\frac{1}{\sin (2 x)}$ .

$\frac{5 \cos (2 x) \cos (5 x)}{\sin (2 x)} - 2 \frac{1^{2}}{\sin^{2} (2 x)} \sin (5 x)$

Step 11.2.6

One to any power is one.

$\frac{5 \cos (2 x) \cos (5 x)}{\sin (2 x)} - 2 \frac{1}{\sin^{2} (2 x)} \sin (5 x)$

Step 11.2.7

Combine $- 2$ and $\frac{1}{\sin^{2} (2 x)}$ .

$\frac{5 \cos (2 x) \cos (5 x)}{\sin (2 x)} + \frac{- 2}{\sin^{2} (2 x)} \sin (5 x)$

Step 11.2.8

Move the negative in front of the fraction.

$\frac{5 \cos (2 x) \cos (5 x)}{\sin (2 x)} - \frac{2}{\sin^{2} (2 x)} \sin (5 x)$

Step 11.2.9

Combine $\sin (5 x)$ and $\frac{2}{\sin^{2} (2 x)}$ .

$\frac{5 \cos (2 x) \cos (5 x)}{\sin (2 x)} - \frac{\sin (5 x) \cdot 2}{\sin^{2} (2 x)}$

Step 11.2.10

Move $2$ to the left of $\sin (5 x)$ .

$\frac{5 \cos (2 x) \cos (5 x)}{\sin (2 x)} - \frac{2 \sin (5 x)}{\sin^{2} (2 x)}$

Step 11.3

Simplify each term.

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Step 11.3.1

Separate fractions.

$\frac{5 \cos (5 x)}{1} \cdot \frac{\cos (2 x)}{\sin (2 x)} - \frac{2 \sin (5 x)}{\sin^{2} (2 x)}$

Step 11.3.2

Convert from $\frac{\cos (2 x)}{\sin (2 x)}$ to $\cot (2 x)$ .

$\frac{5 \cos (5 x)}{1} \cot (2 x) - \frac{2 \sin (5 x)}{\sin^{2} (2 x)}$

Step 11.3.3

Divide $5 \cos (5 x)$ by $1$ .

$5 \cos (5 x) \cot (2 x) - \frac{2 \sin (5 x)}{\sin^{2} (2 x)}$

Step 11.3.4

Factor $\sin (2 x)$ out of $\sin^{2} (2 x)$ .

$5 \cos (5 x) \cot (2 x) - \frac{2 \sin (5 x)}{\sin (2 x) \sin (2 x)}$

Step 11.3.5

Separate fractions.

$5 \cos (5 x) \cot (2 x) - (\frac{2}{\sin (2 x)} \cdot \frac{\sin (5 x)}{\sin (2 x)})$

Step 11.3.6

Rewrite $\frac{\sin (5 x)}{\sin (2 x)}$ as a product.

$5 \cos (5 x) \cot (2 x) - (\frac{2}{\sin (2 x)} (\sin (5 x) \frac{1}{\sin (2 x)}))$

Step 11.3.7

Write $\sin (5 x)$ as a fraction with denominator $1$ .

$5 \cos (5 x) \cot (2 x) - (\frac{2}{\sin (2 x)} (\frac{\sin (5 x)}{1} \cdot \frac{1}{\sin (2 x)}))$

Step 11.3.8

Simplify.

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Step 11.3.8.1

Divide $\sin (5 x)$ by $1$ .

$5 \cos (5 x) \cot (2 x) - (\frac{2}{\sin (2 x)} (\sin (5 x) \frac{1}{\sin (2 x)}))$

Step 11.3.8.2

Convert from $\frac{1}{\sin (2 x)}$ to $\csc (2 x)$ .

$5 \cos (5 x) \cot (2 x) - (\frac{2}{\sin (2 x)} (\sin (5 x) \csc (2 x)))$

Step 11.3.9

Separate fractions.

$5 \cos (5 x) \cot (2 x) - (\frac{2}{1} \cdot \frac{1}{\sin (2 x)} (\sin (5 x) \csc (2 x)))$

Step 11.3.10

Convert from $\frac{1}{\sin (2 x)}$ to $\csc (2 x)$ .

$5 \cos (5 x) \cot (2 x) - (\frac{2}{1} \csc (2 x) (\sin (5 x) \csc (2 x)))$

Step 11.3.11

Divide $2$ by $1$ .

$5 \cos (5 x) \cot (2 x) - (2 \csc (2 x) (\sin (5 x) \csc (2 x)))$

Step 11.3.12

Multiply $2 \csc (2 x) (\sin (5 x) \csc (2 x))$ .

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Step 11.3.12.1

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

$5 \cos (5 x) \cot (2 x) - (2 (\csc^{1} (2 x) \csc (2 x)) \sin (5 x))$

Step 11.3.12.2

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

$5 \cos (5 x) \cot (2 x) - (2 (\csc^{1} (2 x) \csc^{1} (2 x)) \sin (5 x))$

Step 11.3.12.3

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

$5 \cos (5 x) \cot (2 x) - (2 {\csc (2 x)}^{1 + 1} \sin (5 x))$

Step 11.3.12.4

Add $1$ and $1$ .

$5 \cos (5 x) \cot (2 x) - (2 \csc^{2} (2 x) \sin (5 x))$

Step 11.3.13

Multiply $2$ by $- 1$ .

$5 \cos (5 x) \cot (2 x) - 2 \csc^{2} (2 x) \sin (5 x)$