Calculus Examples

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

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

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

Step 2

Differentiate using the Sum Rule.

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Step 2.1

Multiply the exponents in ${(e^{x})}^{2}$ .

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Step 2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.2

Move $2$ to the left of $x$ .

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

Step 2.2

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

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

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u$ as $- x$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $- x$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $- 1$ by $1$ .

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

Step 4.3.2

Move $- 1$ to the left of $e^{- x}$ .

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

Step 4.3.3

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

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

Step 4.4

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

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

Step 4.5

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

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

Step 5

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

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Step 5.1

Move $e^{- x}$ .

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

Step 5.2

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

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

Step 5.3

Add $- x$ and $x$ .

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

Step 6

Simplify $e^{0} \cdot - 1$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Simplify the numerator.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

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

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Step 8.3.1.1.1

Move $e^{x}$ .

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

Step 8.3.1.1.2

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

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

Step 8.3.1.1.3

Subtract $x$ from $x$ .

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

Step 8.3.1.2

Simplify $- e^{0}$ .

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

Step 8.3.1.3

Multiply $- 1$ by $1$ .

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

Step 8.3.1.4

Rewrite $- 1 e^{x}$ as $- e^{x}$ .

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

Step 8.3.2

Subtract $1$ from $- 1$ .

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

Step 8.4

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 8.5

Factor $- 1$ out of $- e^{x}$ .

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

Step 8.6

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

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

Step 8.7

Move the negative in front of the fraction.

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