Calculus Examples

$y = \frac{\cos (m x)}{\sin (n x)}$

Step 1

Rewrite $\frac{\cos (m x)}{\sin (n x)}$ as a product.

$\frac{d}{d x} [\cos (m x) \frac{1}{\sin (n x)}]$

Step 2

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

$\frac{d}{d x} [\frac{\cos (m x)}{1} \cdot \frac{1}{\sin (n x)}]$

Step 3

Simplify.

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Step 3.1

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

$\frac{d}{d x} [\cos (m x) \frac{1}{\sin (n x)}]$

Step 3.2

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

$\frac{d}{d x} [\cos (m x) \csc (n x)]$

Step 4

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 $f (x) = \cos (m x)$ and $g (x) = \csc (n x)$ .

$\cos (m x) \frac{d}{d x} [\csc (n x)] + \csc (n x) \frac{d}{d x} [\cos (m x)]$

Step 5

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 $g (x) = n x$ .

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Step 5.1

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

$\cos (m x) (\frac{d}{d u_{1}} [\csc (u_{1})] \frac{d}{d x} [n x]) + \csc (n x) \frac{d}{d x} [\cos (m x)]$

Step 5.2

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

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

Step 5.3

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

$\cos (m x) (- \csc (n x) \cot (n x) \frac{d}{d x} [n x]) + \csc (n x) \frac{d}{d x} [\cos (m x)]$

Step 6

Differentiate.

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Step 6.1

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

$\cos (m x) (- \csc (n x) \cot (n x) (n \frac{d}{d x} [x])) + \csc (n x) \frac{d}{d x} [\cos (m x)]$

Step 6.2

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

$\cos (m x) (- \csc (n x) \cot (n x) (n \cdot 1)) + \csc (n x) \frac{d}{d x} [\cos (m x)]$

Step 6.3

Multiply $n$ by $1$ .

$\cos (m x) (- \csc (n x) \cot (n x) n) + \csc (n x) \frac{d}{d x} [\cos (m x)]$

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

$\cos (m x) (- \csc (n x) \cot (n x) n) + \csc (n x) (- \sin (m x) \frac{d}{d x} [m x])$

Step 8

Differentiate.

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Step 8.1

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

$\cos (m x) (- \csc (n x) \cot (n x) n) + \csc (n x) (- \sin (m x) (m \frac{d}{d x} [x]))$

Step 8.2

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

$\cos (m x) (- \csc (n x) \cot (n x) n) + \csc (n x) (- \sin (m x) (m \cdot 1))$

Step 8.3

Multiply $m$ by $1$ .

$\cos (m x) (- \csc (n x) \cot (n x) n) + \csc (n x) (- \sin (m x) m)$

Step 9

Simplify.

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Step 9.1

Reorder terms.

$- n \cos (m x) \cot (n x) \csc (n x) - m \csc (n x) \sin (m x)$

Step 9.2

Simplify each term.

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Step 9.2.1

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

$- n \cos (m x) \frac{\cos (n x)}{\sin (n x)} \csc (n x) - m \csc (n x) \sin (m x)$

Step 9.2.2

Multiply $- n \cos (m x) \frac{\cos (n x)}{\sin (n x)}$ .

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Step 9.2.2.1

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

$- \frac{\cos (n x) n}{\sin (n x)} \cos (m x) \csc (n x) - m \csc (n x) \sin (m x)$

Step 9.2.2.2

Combine $\cos (m x)$ and $\frac{\cos (n x) n}{\sin (n x)}$ .

$- \frac{\cos (m x) (\cos (n x) n)}{\sin (n x)} \csc (n x) - m \csc (n x) \sin (m x)$

Step 9.2.3

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

$- \frac{\cos (m x) \cos (n x) n}{\sin (n x)} \cdot \frac{1}{\sin (n x)} - m \csc (n x) \sin (m x)$

Step 9.2.4

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

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Step 9.2.4.1

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

$- \frac{\cos (m x) \cos (n x) n}{\sin (n x) \sin (n x)} - m \csc (n x) \sin (m x)$

Step 9.2.4.2

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

$- \frac{\cos (m x) \cos (n x) n}{\sin^{1} (n x) \sin (n x)} - m \csc (n x) \sin (m x)$

Step 9.2.4.3

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

$- \frac{\cos (m x) \cos (n x) n}{\sin^{1} (n x) \sin^{1} (n x)} - m \csc (n x) \sin (m x)$

Step 9.2.4.4

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

$- \frac{\cos (m x) \cos (n x) n}{{\sin (n x)}^{1 + 1}} - m \csc (n x) \sin (m x)$

Step 9.2.4.5

Add $1$ and $1$ .

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

Step 9.2.5

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

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

Step 9.2.6

Combine $\frac{1}{\sin (n x)}$ and $m$ .

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

Step 9.2.7

Combine $\sin (m x)$ and $\frac{m}{\sin (n x)}$ .

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

Step 9.3

Simplify each term.

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Step 9.3.1

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

$- \frac{\cos (m x) \cos (n x) n}{\sin (n x) \sin (n x)} - \frac{\sin (m x) m}{\sin (n x)}$

Step 9.3.2

Separate fractions.

$- (\frac{\cos (n x) n}{\sin (n x)} \cdot \frac{\cos (m x)}{\sin (n x)}) - \frac{\sin (m x) m}{\sin (n x)}$

Step 9.3.3

Rewrite $\frac{\cos (m x)}{\sin (n x)}$ as a product.

$- (\frac{\cos (n x) n}{\sin (n x)} (\cos (m x) \frac{1}{\sin (n x)})) - \frac{\sin (m x) m}{\sin (n x)}$

Step 9.3.4

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

$- (\frac{\cos (n x) n}{\sin (n x)} (\frac{\cos (m x)}{1} \cdot \frac{1}{\sin (n x)})) - \frac{\sin (m x) m}{\sin (n x)}$

Step 9.3.5

Simplify.

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Step 9.3.5.1

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

$- (\frac{\cos (n x) n}{\sin (n x)} (\cos (m x) \frac{1}{\sin (n x)})) - \frac{\sin (m x) m}{\sin (n x)}$

Step 9.3.5.2

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

$- (\frac{\cos (n x) n}{\sin (n x)} (\cos (m x) \csc (n x))) - \frac{\sin (m x) m}{\sin (n x)}$

Step 9.3.6

Separate fractions.

$- (\frac{n}{1} \cdot \frac{\cos (n x)}{\sin (n x)} (\cos (m x) \csc (n x))) - \frac{\sin (m x) m}{\sin (n x)}$

Step 9.3.7

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

$- (\frac{n}{1} \cot (n x) (\cos (m x) \csc (n x))) - \frac{\sin (m x) m}{\sin (n x)}$

Step 9.3.8

Divide $n$ by $1$ .

$- (n \cot (n x) (\cos (m x) \csc (n x))) - \frac{\sin (m x) m}{\sin (n x)}$

Step 9.3.9

Separate fractions.

$- n \cot (n x) \cos (m x) \csc (n x) - (\frac{m}{1} \cdot \frac{\sin (m x)}{\sin (n x)})$

Step 9.3.10

Rewrite $\frac{\sin (m x)}{\sin (n x)}$ as a product.

$- n \cot (n x) \cos (m x) \csc (n x) - (\frac{m}{1} (\sin (m x) \frac{1}{\sin (n x)}))$

Step 9.3.11

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

$- n \cot (n x) \cos (m x) \csc (n x) - (\frac{m}{1} (\frac{\sin (m x)}{1} \cdot \frac{1}{\sin (n x)}))$

Step 9.3.12

Simplify.

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Step 9.3.12.1

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

$- n \cot (n x) \cos (m x) \csc (n x) - (\frac{m}{1} (\sin (m x) \frac{1}{\sin (n x)}))$

Step 9.3.12.2

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

$- n \cot (n x) \cos (m x) \csc (n x) - (\frac{m}{1} (\sin (m x) \csc (n x)))$

Step 9.3.13

Divide $m$ by $1$ .

$- n \cot (n x) \cos (m x) \csc (n x) - m (\sin (m x) \csc (n x))$

Step 9.4

Reorder factors in $- n \cot (n x) \cos (m x) \csc (n x) - m (\sin (m x) \csc (n x))$ .

$- n \cot (n x) \cos (m x) \csc (n x) - m \sin (m x) \csc (n x)$