Calculus Examples

$y = \frac{e^{x}}{1 - e^{x}}$

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 $f (x) = e^{x}$ and $g (x) = 1 - e^{x}$ .

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

Step 2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Add $0$ and $\frac{d}{d x} [- e^{x}]$ .

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

Step 3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- e^{x}$ with respect to $x$ is $- \frac{d}{d x} [e^{x}]$ .

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

Step 3.5

Multiply.

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Step 3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.2

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

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

Step 4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Add $x$ and $x$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Simplify the numerator.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

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

Step 6.2.1.2

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

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Step 6.2.1.2.1

Move $e^{x}$ .

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

Step 6.2.1.2.2

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

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

Step 6.2.1.2.3

Add $x$ and $x$ .

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

Step 6.2.2

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

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Step 6.2.2.1

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

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

Step 6.2.2.2

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

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