Calculus Examples

$y = \frac{\sinh (2 x)}{\cosh (2 x) - 5}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \sinh (2 x)$ and $g (x) = \cosh (2 x) - 5$ .

$\frac{(\cosh (2 x) - 5) \frac{d}{d x} [\sinh (2 x)] - \sinh (2 x) \frac{d}{d x} [\cosh (2 x) - 5]}{{(\cosh (2 x) - 5)}^{2}}$

Step 2

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

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Step 2.1

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

$\frac{(\cosh (2 x) - 5) (\frac{d}{d u_{1}} [\sinh (u_{1})] \frac{d}{d x} [2 x]) - \sinh (2 x) \frac{d}{d x} [\cosh (2 x) - 5]}{{(\cosh (2 x) - 5)}^{2}}$

Step 2.2

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

$\frac{(\cosh (2 x) - 5) (\cosh (u_{1}) \frac{d}{d x} [2 x]) - \sinh (2 x) \frac{d}{d x} [\cosh (2 x) - 5]}{{(\cosh (2 x) - 5)}^{2}}$

Step 2.3

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

$\frac{(\cosh (2 x) - 5) (\cosh (2 x) \frac{d}{d x} [2 x]) - \sinh (2 x) \frac{d}{d x} [\cosh (2 x) - 5]}{{(\cosh (2 x) - 5)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$\frac{(\cosh (2 x) - 5) \cosh (2 x) (2 \frac{d}{d x} [x]) - \sinh (2 x) \frac{d}{d x} [\cosh (2 x) - 5]}{{(\cosh (2 x) - 5)}^{2}}$

Step 3.2

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

$\frac{(\cosh (2 x) - 5) \cosh (2 x) (2 \cdot 1) - \sinh (2 x) \frac{d}{d x} [\cosh (2 x) - 5]}{{(\cosh (2 x) - 5)}^{2}}$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $2$ by $1$ .

$\frac{(\cosh (2 x) - 5) \cosh (2 x) \cdot 2 - \sinh (2 x) \frac{d}{d x} [\cosh (2 x) - 5]}{{(\cosh (2 x) - 5)}^{2}}$

Step 3.3.2

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

$\frac{2 \cdot ((\cosh (2 x) - 5) \cosh (2 x)) - \sinh (2 x) \frac{d}{d x} [\cosh (2 x) - 5]}{{(\cosh (2 x) - 5)}^{2}}$

Step 3.4

By the Sum Rule, the derivative of $\cosh (2 x) - 5$ with respect to $x$ is $\frac{d}{d x} [\cosh (2 x)] + \frac{d}{d x} [- 5]$ .

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - \sinh (2 x) (\frac{d}{d x} [\cosh (2 x)] + \frac{d}{d x} [- 5])}{{(\cosh (2 x) - 5)}^{2}}$

Step 4

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

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Step 4.1

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

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - \sinh (2 x) (\frac{d}{d u_{2}} [\cosh (u_{2})] \frac{d}{d x} [2 x] + \frac{d}{d x} [- 5])}{{(\cosh (2 x) - 5)}^{2}}$

Step 4.2

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

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - \sinh (2 x) (\sinh (u_{2}) \frac{d}{d x} [2 x] + \frac{d}{d x} [- 5])}{{(\cosh (2 x) - 5)}^{2}}$

Step 4.3

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

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - \sinh (2 x) (\sinh (2 x) \frac{d}{d x} [2 x] + \frac{d}{d x} [- 5])}{{(\cosh (2 x) - 5)}^{2}}$

Step 5

Differentiate.

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Step 5.1

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

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - \sinh (2 x) (\sinh (2 x) (2 \frac{d}{d x} [x]) + \frac{d}{d x} [- 5])}{{(\cosh (2 x) - 5)}^{2}}$

Step 5.2

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

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - \sinh (2 x) (\sinh (2 x) (2 \cdot 1) + \frac{d}{d x} [- 5])}{{(\cosh (2 x) - 5)}^{2}}$

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply $2$ by $1$ .

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - \sinh (2 x) (\sinh (2 x) \cdot 2 + \frac{d}{d x} [- 5])}{{(\cosh (2 x) - 5)}^{2}}$

Step 5.3.2

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

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - \sinh (2 x) (2 \cdot \sinh (2 x) + \frac{d}{d x} [- 5])}{{(\cosh (2 x) - 5)}^{2}}$

Step 5.4

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

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - \sinh (2 x) (2 \sinh (2 x) + 0)}{{(\cosh (2 x) - 5)}^{2}}$

Step 5.5

Simplify the expression.

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Step 5.5.1

Add $2 \sinh (2 x)$ and $0$ .

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - \sinh (2 x) (2 \sinh (2 x))}{{(\cosh (2 x) - 5)}^{2}}$

Step 5.5.2

Multiply $2$ by $- 1$ .

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - 2 \sinh (2 x) \sinh (2 x)}{{(\cosh (2 x) - 5)}^{2}}$

Step 6

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

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - 2 (\sinh^{1} (2 x) \sinh (2 x))}{{(\cosh (2 x) - 5)}^{2}}$

Step 7

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

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - 2 (\sinh^{1} (2 x) \sinh^{1} (2 x))}{{(\cosh (2 x) - 5)}^{2}}$

Step 8

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

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - 2 {\sinh (2 x)}^{1 + 1}}{{(\cosh (2 x) - 5)}^{2}}$

Step 9

Add $1$ and $1$ .

$\frac{2 (\cosh (2 x) - 5) \cosh (2 x) - 2 \sinh^{2} (2 x)}{{(\cosh (2 x) - 5)}^{2}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{(2 \cosh (2 x) + 2 \cdot - 5) \cosh (2 x) - 2 \sinh^{2} (2 x)}{{(\cosh (2 x) - 5)}^{2}}$

Step 10.2

Apply the distributive property.

$\frac{2 \cosh (2 x) \cosh (2 x) + 2 \cdot - 5 \cosh (2 x) - 2 \sinh^{2} (2 x)}{{(\cosh (2 x) - 5)}^{2}}$

Step 10.3

Simplify each term.

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Step 10.3.1

Multiply $2 \cosh (2 x) \cosh (2 x)$ .

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Step 10.3.1.1

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

$\frac{2 (\cosh^{1} (2 x) \cosh (2 x)) + 2 \cdot - 5 \cosh (2 x) - 2 \sinh^{2} (2 x)}{{(\cosh (2 x) - 5)}^{2}}$

Step 10.3.1.2

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

$\frac{2 (\cosh^{1} (2 x) \cosh^{1} (2 x)) + 2 \cdot - 5 \cosh (2 x) - 2 \sinh^{2} (2 x)}{{(\cosh (2 x) - 5)}^{2}}$

Step 10.3.1.3

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

$\frac{2 {\cosh (2 x)}^{1 + 1} + 2 \cdot - 5 \cosh (2 x) - 2 \sinh^{2} (2 x)}{{(\cosh (2 x) - 5)}^{2}}$

Step 10.3.1.4

Add $1$ and $1$ .

$\frac{2 \cosh^{2} (2 x) + 2 \cdot - 5 \cosh (2 x) - 2 \sinh^{2} (2 x)}{{(\cosh (2 x) - 5)}^{2}}$

Step 10.3.2

Multiply $2$ by $- 5$ .

$\frac{2 \cosh^{2} (2 x) - 10 \cosh (2 x) - 2 \sinh^{2} (2 x)}{{(\cosh (2 x) - 5)}^{2}}$

Step 10.4

Reorder terms.

$\frac{2 \cosh^{2} (2 x) - 2 \sinh^{2} (2 x) - 10 \cosh (2 x)}{{(\cosh (2 x) - 5)}^{2}}$