Calculus Examples

$f (x) = \ln (\frac{e^{x}}{e^{x} + 1})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{e^{x}}{e^{x} + 1}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{e^{x}}{e^{x} + 1}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{e^{x}}{e^{x} + 1}]$

Step 1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [\frac{e^{x}}{e^{x} + 1}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{e^{x}}{e^{x} + 1}$ .

$\frac{1}{\frac{e^{x}}{e^{x} + 1}} \frac{d}{d x} [\frac{e^{x}}{e^{x} + 1}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{e^{x}}{e^{x} + 1}$ .

$1 \frac{e^{x} + 1}{e^{x}} \frac{d}{d x} [\frac{e^{x}}{e^{x} + 1}]$

Step 3

Multiply $\frac{e^{x} + 1}{e^{x}}$ by $1$ .

$\frac{e^{x} + 1}{e^{x}} \frac{d}{d x} [\frac{e^{x}}{e^{x} + 1}]$

Step 4

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 $g (x) = e^{x} + 1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Differentiate using the Constant Rule.

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Step 8.1

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

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

Step 8.2

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

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

Step 9

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

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Step 9.1

Move $e^{x}$ .

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

Step 9.2

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

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

Step 9.3

Add $x$ and $x$ .

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

Step 10

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

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

Step 11

Cancel the common factors.

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Step 11.1

Factor $e^{x} + 1$ out of $e^{x} {(e^{x} + 1)}^{2}$ .

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

Step 11.2

Cancel the common factor.

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

Step 11.3

Rewrite the expression.

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

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Simplify the numerator.

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Step 12.3.1

Simplify each term.

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Step 12.3.1.1

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

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Step 12.3.1.1.1

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

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

Step 12.3.1.1.2

Add $x$ and $x$ .

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

Step 12.3.1.2

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

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

Step 12.3.2

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

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Step 12.3.2.1

Subtract $e^{2 x}$ from $e^{2 x}$ .

$\frac{0 + e^{x}}{e^{x} e^{x} + e^{x} \cdot 1}$

Step 12.3.2.2

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

$\frac{e^{x}}{e^{x} e^{x} + e^{x} \cdot 1}$

Step 12.4

Combine terms.

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Step 12.4.1

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

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Step 12.4.1.1

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

$\frac{e^{x}}{e^{x + x} + e^{x} \cdot 1}$

Step 12.4.1.2

Add $x$ and $x$ .

$\frac{e^{x}}{e^{2 x} + e^{x} \cdot 1}$

Step 12.4.2

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

$\frac{e^{x}}{e^{2 x} + e^{x}}$

Step 12.5

Simplify the denominator.

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Step 12.5.1

Rewrite $e^{2 x}$ as ${(e^{x})}^{2}$ .

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

Step 12.5.2

Let $u = e^{x}$ . Substitute $u$ for all occurrences of $e^{x}$ .

$\frac{e^{x}}{u^{2} + u}$

Step 12.5.3

Factor $u$ out of $u^{2} + u$ .

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Step 12.5.3.1

Factor $u$ out of $u^{2}$ .

$\frac{e^{x}}{u \cdot u + u}$

Step 12.5.3.2

Raise $u$ to the power of $1$ .

$\frac{e^{x}}{u \cdot u + u^{1}}$

Step 12.5.3.3

Factor $u$ out of $u^{1}$ .

$\frac{e^{x}}{u \cdot u + u \cdot 1}$

Step 12.5.3.4

Factor $u$ out of $u \cdot u + u \cdot 1$ .

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

Step 12.5.4

Replace all occurrences of $u$ with $e^{x}$ .

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

Step 12.6

Cancel the common factor of $e^{x}$ .

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Step 12.6.1

Cancel the common factor.

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

Step 12.6.2

Rewrite the expression.

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