Calculus Examples

$\frac{f (x) (\sin (14 x))}{\sin (7 x)}$

Step 1

Since $f (x)$ is constant with respect to $x$ , the derivative of $\frac{f (x) \sin (14 x)}{\sin (7 x)}$ with respect to $x$ is $f (x) \frac{d}{d x} [\frac{\sin (14 x)}{\sin (7 x)}]$ .

$f (x) \frac{d}{d x} [\frac{\sin (14 x)}{\sin (7 x)}]$

Step 2

Rewrite $\frac{\sin (14 x)}{\sin (7 x)}$ as a product.

$f (x) \frac{d}{d x} [\sin (14 x) \frac{1}{\sin (7 x)}]$

Step 3

Write $\sin (14 x)$ as a fraction with denominator $1$ .

$f (x) \frac{d}{d x} [\frac{\sin (14 x)}{1} \cdot \frac{1}{\sin (7 x)}]$

Step 4

Simplify.

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Step 4.1

Divide $\sin (14 x)$ by $1$ .

$f (x) \frac{d}{d x} [\sin (14 x) \frac{1}{\sin (7 x)}]$

Step 4.2

Convert from $\frac{1}{\sin (7 x)}$ to $\csc (7 x)$ .

$f (x) \frac{d}{d x} [\sin (14 x) \csc (7 x)]$

Step 5

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 $f (x) = \sin (14 x)$ and $g (x) = \csc (7 x)$ .

$f (x) (\sin (14 x) \frac{d}{d x} [\csc (7 x)] + \csc (7 x) \frac{d}{d x} [\sin (14 x)])$

Step 6

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 $g (x) = 7 x$ .

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Step 6.1

To apply the Chain Rule, set $u_{1}$ as $7 x$ .

$f (x) (\sin (14 x) (\frac{d}{d u_{1}} [\csc (u_{1})] \frac{d}{d x} [7 x]) + \csc (7 x) \frac{d}{d x} [\sin (14 x)])$

Step 6.2

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

$f (x) (\sin (14 x) (- \csc (u_{1}) \cot (u_{1}) \frac{d}{d x} [7 x]) + \csc (7 x) \frac{d}{d x} [\sin (14 x)])$

Step 6.3

Replace all occurrences of $u_{1}$ with $7 x$ .

$f (x) (\sin (14 x) (- \csc (7 x) \cot (7 x) \frac{d}{d x} [7 x]) + \csc (7 x) \frac{d}{d x} [\sin (14 x)])$

Step 7

Differentiate.

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Step 7.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$f (x) (\sin (14 x) (- \csc (7 x) \cot (7 x) (7 \frac{d}{d x} [x])) + \csc (7 x) \frac{d}{d x} [\sin (14 x)])$

Step 7.2

Multiply $7$ by $- 1$ .

$f (x) (\sin (14 x) (- 7 \csc (7 x) \cot (7 x) \frac{d}{d x} [x]) + \csc (7 x) \frac{d}{d x} [\sin (14 x)])$

Step 7.3

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

$f (x) (\sin (14 x) (- 7 \csc (7 x) \cot (7 x) \cdot 1) + \csc (7 x) \frac{d}{d x} [\sin (14 x)])$

Step 7.4

Multiply $- 7$ by $1$ .

$f (x) (\sin (14 x) (- 7 \csc (7 x) \cot (7 x)) + \csc (7 x) \frac{d}{d x} [\sin (14 x)])$

Step 8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 14 x$ .

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Step 8.1

To apply the Chain Rule, set $u_{2}$ as $14 x$ .

$f (x) (\sin (14 x) (- 7 \csc (7 x) \cot (7 x)) + \csc (7 x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [14 x]))$

Step 8.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$f (x) (\sin (14 x) (- 7 \csc (7 x) \cot (7 x)) + \csc (7 x) (\cos (u_{2}) \frac{d}{d x} [14 x]))$

Step 8.3

Replace all occurrences of $u_{2}$ with $14 x$ .

$f (x) (\sin (14 x) (- 7 \csc (7 x) \cot (7 x)) + \csc (7 x) (\cos (14 x) \frac{d}{d x} [14 x]))$

Step 9

Differentiate.

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Step 9.1

Since $14$ is constant with respect to $x$ , the derivative of $14 x$ with respect to $x$ is $14 \frac{d}{d x} [x]$ .

$f (x) (\sin (14 x) (- 7 \csc (7 x) \cot (7 x)) + \csc (7 x) \cos (14 x) (14 \frac{d}{d x} [x]))$

Step 9.2

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

$f (x) (\sin (14 x) (- 7 \csc (7 x) \cot (7 x)) + \csc (7 x) \cos (14 x) (14 \cdot 1))$

Step 9.3

Simplify the expression.

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Step 9.3.1

Multiply $14$ by $1$ .

$f (x) (\sin (14 x) (- 7 \csc (7 x) \cot (7 x)) + \csc (7 x) \cos (14 x) \cdot 14)$

Step 9.3.2

Move $14$ to the left of $\csc (7 x) \cos (14 x)$ .

$f (x) (\sin (14 x) (- 7 \csc (7 x) \cot (7 x)) + 14 \cdot (\csc (7 x) \cos (14 x)))$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$f (x) (\sin (14 x) (- 7 \csc (7 x) \cot (7 x))) + f (x) (14 \csc (7 x) \cos (14 x))$

Step 10.2

Move $14$ to the left of $f (x)$ .

$f (x) \sin (14 x) (- 7 \csc (7 x) \cot (7 x)) + 14 f (x) \csc (7 x) \cos (14 x)$

Step 10.3

Reorder terms.

$- 7 \cot (7 x) \csc (7 x) \sin (14 x) f (x) + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4

Simplify each term.

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Step 10.4.1

Rewrite $\cot (7 x)$ in terms of sines and cosines.

$- 7 \frac{\cos (7 x)}{\sin (7 x)} \csc (7 x) \sin (14 x) f (x) + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.2

Combine $- 7$ and $\frac{\cos (7 x)}{\sin (7 x)}$ .

$\frac{- 7 \cos (7 x)}{\sin (7 x)} \csc (7 x) \sin (14 x) f (x) + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.3

Move the negative in front of the fraction.

$- \frac{(7) \cos (7 x)}{\sin (7 x)} \csc (7 x) \sin (14 x) f (x) + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.4

Rewrite $\csc (7 x)$ in terms of sines and cosines.

$- \frac{7 \cos (7 x)}{\sin (7 x)} \cdot \frac{1}{\sin (7 x)} \sin (14 x) f (x) + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.5

Multiply $- \frac{7 \cos (7 x)}{\sin (7 x)} \cdot \frac{1}{\sin (7 x)}$ .

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Step 10.4.5.1

Multiply $\frac{1}{\sin (7 x)}$ by $\frac{7 \cos (7 x)}{\sin (7 x)}$ .

$- \frac{7 \cos (7 x)}{\sin (7 x) \sin (7 x)} \sin (14 x) f (x) + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.5.2

Raise $\sin (7 x)$ to the power of $1$ .

$- \frac{7 \cos (7 x)}{\sin^{1} (7 x) \sin (7 x)} \sin (14 x) f (x) + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.5.3

Raise $\sin (7 x)$ to the power of $1$ .

$- \frac{7 \cos (7 x)}{\sin^{1} (7 x) \sin^{1} (7 x)} \sin (14 x) f (x) + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.5.4

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

$- \frac{7 \cos (7 x)}{{\sin (7 x)}^{1 + 1}} \sin (14 x) f (x) + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.5.5

Add $1$ and $1$ .

$- \frac{7 \cos (7 x)}{\sin^{2} (7 x)} \sin (14 x) f (x) + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.6

Combine $\sin (14 x)$ and $\frac{7 \cos (7 x)}{\sin^{2} (7 x)}$ .

$- \frac{\sin (14 x) (7 \cos (7 x))}{\sin^{2} (7 x)} f (x) + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.7

Move $7$ to the left of $\sin (14 x)$ .

$- \frac{7 \cdot \sin (14 x) \cos (7 x)}{\sin^{2} (7 x)} f (x) + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.8

Combine $f (x)$ and $\frac{7 \sin (14 x) \cos (7 x)}{\sin^{2} (7 x)}$ .

$- \frac{f (x) (7 \sin (14 x) \cos (7 x))}{\sin^{2} (7 x)} + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.9

Move $7$ to the left of $f (x)$ .

$- \frac{7 \cdot f (x) \sin (14 x) \cos (7 x)}{\sin^{2} (7 x)} + 14 \cos (14 x) \csc (7 x) f (x)$

Step 10.4.10

Rewrite $\csc (7 x)$ in terms of sines and cosines.

$- \frac{7 f (x) \sin (14 x) \cos (7 x)}{\sin^{2} (7 x)} + 14 \cos (14 x) \frac{1}{\sin (7 x)} f (x)$

Step 10.4.11

Multiply $14 \cos (14 x) \frac{1}{\sin (7 x)}$ .

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Step 10.4.11.1

Combine $\frac{1}{\sin (7 x)}$ and $14$ .

$- \frac{7 f (x) \sin (14 x) \cos (7 x)}{\sin^{2} (7 x)} + \frac{14}{\sin (7 x)} \cos (14 x) f (x)$

Step 10.4.11.2

Combine $\frac{14}{\sin (7 x)}$ and $\cos (14 x)$ .

$- \frac{7 f (x) \sin (14 x) \cos (7 x)}{\sin^{2} (7 x)} + \frac{14 \cos (14 x)}{\sin (7 x)} f (x)$

Step 10.4.12

Combine $\frac{14 \cos (14 x)}{\sin (7 x)}$ and $f (x)$ .

$- \frac{7 f (x) \sin (14 x) \cos (7 x)}{\sin^{2} (7 x)} + \frac{14 \cos (14 x) f (x)}{\sin (7 x)}$

Step 10.5

Simplify each term.

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Step 10.5.1

Factor $\sin (7 x)$ out of $\sin^{2} (7 x)$ .

$- \frac{7 f (x) \sin (14 x) \cos (7 x)}{\sin (7 x) \sin (7 x)} + \frac{14 \cos (14 x) f (x)}{\sin (7 x)}$

Step 10.5.2

Separate fractions.

$- (\frac{7 f (x) \cos (7 x)}{\sin (7 x)} \cdot \frac{\sin (14 x)}{\sin (7 x)}) + \frac{14 \cos (14 x) f (x)}{\sin (7 x)}$

Step 10.5.3

Rewrite $\frac{\sin (14 x)}{\sin (7 x)}$ as a product.

$- (\frac{7 f (x) \cos (7 x)}{\sin (7 x)} (\sin (14 x) \frac{1}{\sin (7 x)})) + \frac{14 \cos (14 x) f (x)}{\sin (7 x)}$

Step 10.5.4

Write $\sin (14 x)$ as a fraction with denominator $1$ .

$- (\frac{7 f (x) \cos (7 x)}{\sin (7 x)} (\frac{\sin (14 x)}{1} \cdot \frac{1}{\sin (7 x)})) + \frac{14 \cos (14 x) f (x)}{\sin (7 x)}$

Step 10.5.5

Simplify.

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Step 10.5.5.1

Divide $\sin (14 x)$ by $1$ .

$- (\frac{7 f (x) \cos (7 x)}{\sin (7 x)} (\sin (14 x) \frac{1}{\sin (7 x)})) + \frac{14 \cos (14 x) f (x)}{\sin (7 x)}$

Step 10.5.5.2

Convert from $\frac{1}{\sin (7 x)}$ to $\csc (7 x)$ .

$- (\frac{7 f (x) \cos (7 x)}{\sin (7 x)} (\sin (14 x) \csc (7 x))) + \frac{14 \cos (14 x) f (x)}{\sin (7 x)}$

Step 10.5.6

Separate fractions.

$- (\frac{7 f (x)}{1} \cdot \frac{\cos (7 x)}{\sin (7 x)} (\sin (14 x) \csc (7 x))) + \frac{14 \cos (14 x) f (x)}{\sin (7 x)}$

Step 10.5.7

Convert from $\frac{\cos (7 x)}{\sin (7 x)}$ to $\cot (7 x)$ .

$- (\frac{7 f (x)}{1} \cot (7 x) (\sin (14 x) \csc (7 x))) + \frac{14 \cos (14 x) f (x)}{\sin (7 x)}$

Step 10.5.8

Divide $7 f (x)$ by $1$ .

$- (7 f (x) \cot (7 x) (\sin (14 x) \csc (7 x))) + \frac{14 \cos (14 x) f (x)}{\sin (7 x)}$

Step 10.5.9

Multiply $7$ by $- 1$ .

$- 7 (f (x) \cot (7 x) \sin (14 x) \csc (7 x)) + \frac{14 \cos (14 x) f (x)}{\sin (7 x)}$

Step 10.5.10

Separate fractions.

$- 7 f (x) \cot (7 x) \sin (14 x) \csc (7 x) + \frac{14 f (x)}{1} \cdot \frac{\cos (14 x)}{\sin (7 x)}$

Step 10.5.11

Rewrite $\frac{\cos (14 x)}{\sin (7 x)}$ as a product.

$- 7 f (x) \cot (7 x) \sin (14 x) \csc (7 x) + \frac{14 f (x)}{1} (\cos (14 x) \frac{1}{\sin (7 x)})$

Step 10.5.12

Write $\cos (14 x)$ as a fraction with denominator $1$ .

$- 7 f (x) \cot (7 x) \sin (14 x) \csc (7 x) + \frac{14 f (x)}{1} (\frac{\cos (14 x)}{1} \cdot \frac{1}{\sin (7 x)})$

Step 10.5.13

Simplify.

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Step 10.5.13.1

Divide $\cos (14 x)$ by $1$ .

$- 7 f (x) \cot (7 x) \sin (14 x) \csc (7 x) + \frac{14 f (x)}{1} (\cos (14 x) \frac{1}{\sin (7 x)})$

Step 10.5.13.2

Convert from $\frac{1}{\sin (7 x)}$ to $\csc (7 x)$ .

$- 7 f (x) \cot (7 x) \sin (14 x) \csc (7 x) + \frac{14 f (x)}{1} (\cos (14 x) \csc (7 x))$

Step 10.5.14

Divide $14 f (x)$ by $1$ .

$- 7 f (x) \cot (7 x) \sin (14 x) \csc (7 x) + 14 f (x) \cos (14 x) \csc (7 x)$

Step 10.6

Reorder factors in $- 7 f (x) \cot (7 x) \sin (14 x) \csc (7 x) + 14 f (x) \cos (14 x) \csc (7 x)$ .

$- 7 \cot (7 x) \sin (14 x) \csc (7 x) f (x) + 14 \cos (14 x) \csc (7 x) f (x)$