Calculus Examples

$\frac{1 + c e^{x}}{1 - c e^{x}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d c} [\frac{f (c)}{g (c)}]$ is $\frac{g (c) \frac{d}{d c} [f (c)] - f (c) \frac{d}{d c} [g (c)]}{{g (c)}^{2}}$ where $f (c) = 1 + c e^{x}$ and $g (c) = 1 - c e^{x}$ .

$\frac{(1 - c e^{x}) \frac{d}{d c} [1 + c e^{x}] - (1 + c e^{x}) \frac{d}{d c} [1 - c e^{x}]}{{(1 - c e^{x})}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $1 + c e^{x}$ with respect to $c$ is $\frac{d}{d c} [1] + \frac{d}{d c} [c e^{x}]$ .

$\frac{(1 - c e^{x}) (\frac{d}{d c} [1] + \frac{d}{d c} [c e^{x}]) - (1 + c e^{x}) \frac{d}{d c} [1 - c e^{x}]}{{(1 - c e^{x})}^{2}}$

Step 2.2

Since $1$ is constant with respect to $c$ , the derivative of $1$ with respect to $c$ is $0$ .

$\frac{(1 - c e^{x}) (0 + \frac{d}{d c} [c e^{x}]) - (1 + c e^{x}) \frac{d}{d c} [1 - c e^{x}]}{{(1 - c e^{x})}^{2}}$

Step 3

Evaluate $\frac{d}{d c} [c e^{x}]$ .

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Step 3.1

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

$\frac{(1 - c e^{x}) (0 + e^{x} \frac{d}{d c} [c]) - (1 + c e^{x}) \frac{d}{d c} [1 - c e^{x}]}{{(1 - c e^{x})}^{2}}$

Step 3.2

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

$\frac{(1 - c e^{x}) (0 + e^{x} \cdot 1) - (1 + c e^{x}) \frac{d}{d c} [1 - c e^{x}]}{{(1 - c e^{x})}^{2}}$

Step 3.3

Multiply $e^{x}$ by $1$ .

$\frac{(1 - c e^{x}) (0 + e^{x}) - (1 + c e^{x}) \frac{d}{d c} [1 - c e^{x}]}{{(1 - c e^{x})}^{2}}$

Step 4

Differentiate.

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Step 4.1

Add $0$ and $e^{x}$ .

$\frac{(1 - c e^{x}) e^{x} - (1 + c e^{x}) \frac{d}{d c} [1 - c e^{x}]}{{(1 - c e^{x})}^{2}}$

Step 4.2

By the Sum Rule, the derivative of $1 - c e^{x}$ with respect to $c$ is $\frac{d}{d c} [1] + \frac{d}{d c} [- c e^{x}]$ .

$\frac{(1 - c e^{x}) e^{x} - (1 + c e^{x}) (\frac{d}{d c} [1] + \frac{d}{d c} [- c e^{x}])}{{(1 - c e^{x})}^{2}}$

Step 4.3

Since $1$ is constant with respect to $c$ , the derivative of $1$ with respect to $c$ is $0$ .

$\frac{(1 - c e^{x}) e^{x} - (1 + c e^{x}) (0 + \frac{d}{d c} [- c e^{x}])}{{(1 - c e^{x})}^{2}}$

Step 5

Evaluate $\frac{d}{d c} [- c e^{x}]$ .

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Step 5.1

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

$\frac{(1 - c e^{x}) e^{x} - (1 + c e^{x}) (0 - e^{x} \frac{d}{d c} [c])}{{(1 - c e^{x})}^{2}}$

Step 5.2

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

$\frac{(1 - c e^{x}) e^{x} - (1 + c e^{x}) (0 - e^{x} \cdot 1)}{{(1 - c e^{x})}^{2}}$

Step 5.3

Multiply $- 1$ by $1$ .

$\frac{(1 - c e^{x}) e^{x} - (1 + c e^{x}) (0 - e^{x})}{{(1 - c e^{x})}^{2}}$

Step 6

Subtract $e^{x}$ from $0$ .

$\frac{(1 - c e^{x}) e^{x} - (1 + c e^{x}) (- e^{x})}{{(1 - c e^{x})}^{2}}$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{1 e^{x} - c e^{x} e^{x} - (1 + c e^{x}) (- e^{x})}{{(1 - c e^{x})}^{2}}$

Step 7.2

Apply the distributive property.

$\frac{1 e^{x} - c e^{x} e^{x} + (- 1 \cdot 1 - (c e^{x})) (- e^{x})}{{(1 - c e^{x})}^{2}}$

Step 7.3

Apply the distributive property.

$\frac{1 e^{x} - c e^{x} e^{x} - 1 \cdot 1 (- e^{x}) - (c e^{x}) (- e^{x})}{{(1 - c e^{x})}^{2}}$

Step 7.4

Simplify the numerator.

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Step 7.4.1

Simplify each term.

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Step 7.4.1.1

Multiply $e^{x}$ by $1$ .

$\frac{e^{x} - c e^{x} e^{x} - 1 \cdot 1 (- e^{x}) - (c e^{x}) (- e^{x})}{{(1 - c e^{x})}^{2}}$

Step 7.4.1.2

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 7.4.1.2.1

Move $e^{x}$ .

$\frac{e^{x} - c (e^{x} e^{x}) - 1 \cdot 1 (- e^{x}) - (c e^{x}) (- e^{x})}{{(1 - c e^{x})}^{2}}$

Step 7.4.1.2.2

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

$\frac{e^{x} - c e^{x + x} - 1 \cdot 1 (- e^{x}) - (c e^{x}) (- e^{x})}{{(1 - c e^{x})}^{2}}$

Step 7.4.1.2.3

Add $x$ and $x$ .

$\frac{e^{x} - c e^{2 x} - 1 \cdot 1 (- e^{x}) - (c e^{x}) (- e^{x})}{{(1 - c e^{x})}^{2}}$

Step 7.4.1.3

Multiply $- 1$ by $1$ .

$\frac{e^{x} - c e^{2 x} - 1 (- e^{x}) - (c e^{x}) (- e^{x})}{{(1 - c e^{x})}^{2}}$

Step 7.4.1.4

Multiply $- 1 (- e^{x})$ .

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Step 7.4.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{e^{x} - c e^{2 x} + 1 e^{x} - (c e^{x}) (- e^{x})}{{(1 - c e^{x})}^{2}}$

Step 7.4.1.4.2

Multiply $e^{x}$ by $1$ .

$\frac{e^{x} - c e^{2 x} + e^{x} - (c e^{x}) (- e^{x})}{{(1 - c e^{x})}^{2}}$

Step 7.4.1.5

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 7.4.1.5.1

Move $e^{x}$ .

$\frac{e^{x} - c e^{2 x} + e^{x} - (c (e^{x} e^{x})) \cdot - 1}{{(1 - c e^{x})}^{2}}$

Step 7.4.1.5.2

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

$\frac{e^{x} - c e^{2 x} + e^{x} - (c e^{x + x}) \cdot - 1}{{(1 - c e^{x})}^{2}}$

Step 7.4.1.5.3

Add $x$ and $x$ .

$\frac{e^{x} - c e^{2 x} + e^{x} - (c e^{2 x}) \cdot - 1}{{(1 - c e^{x})}^{2}}$

Step 7.4.1.6

Multiply $- c e^{2 x} \cdot - 1$ .

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Step 7.4.1.6.1

Multiply $- 1$ by $- 1$ .

$\frac{e^{x} - c e^{2 x} + e^{x} + 1 c e^{2 x}}{{(1 - c e^{x})}^{2}}$

Step 7.4.1.6.2

Multiply $c$ by $1$ .

$\frac{e^{x} - c e^{2 x} + e^{x} + c e^{2 x}}{{(1 - c e^{x})}^{2}}$

Step 7.4.2

Combine the opposite terms in $e^{x} - c e^{2 x} + e^{x} + c e^{2 x}$ .

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Step 7.4.2.1

Add $- c e^{2 x}$ and $c e^{2 x}$ .

$\frac{e^{x} + 0 + e^{x}}{{(1 - c e^{x})}^{2}}$

Step 7.4.2.2

Add $e^{x}$ and $0$ .

$\frac{e^{x} + e^{x}}{{(1 - c e^{x})}^{2}}$

Step 7.4.3

Add $e^{x}$ and $e^{x}$ .

$\frac{2 e^{x}}{{(1 - c e^{x})}^{2}}$

Step 7.5

Reorder terms.

$\frac{2 e^{x}}{{(- e^{x} c + 1)}^{2}}$

Step 7.6

Reorder factors in $\frac{2 e^{x}}{{(- e^{x} c + 1)}^{2}}$ .

$\frac{2 e^{x}}{{(- c e^{x} + 1)}^{2}}$