Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

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

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

Step 4

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

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Simplify the numerator.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Expand $(x + e^{x}) (- x e^{x} - e^{x})$ using the FOIL Method.

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Step 7.3.1.1.1

Apply the distributive property.

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

Step 7.3.1.1.2

Apply the distributive property.

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

Step 7.3.1.1.3

Apply the distributive property.

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

Step 7.3.1.2

Simplify each term.

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Step 7.3.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 7.3.1.2.2

Multiply $x$ by $x$ by adding the exponents.

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Step 7.3.1.2.2.1

Move $x$ .

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

Step 7.3.1.2.2.2

Multiply $x$ by $x$ .

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

Step 7.3.1.2.3

Rewrite using the commutative property of multiplication.

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

Step 7.3.1.2.4

Rewrite using the commutative property of multiplication.

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

Step 7.3.1.2.5

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

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Step 7.3.1.2.5.1

Move $e^{x}$ .

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

Step 7.3.1.2.5.2

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

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

Step 7.3.1.2.5.3

Add $x$ and $x$ .

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

Step 7.3.1.2.6

Rewrite using the commutative property of multiplication.

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

Step 7.3.1.2.7

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

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Step 7.3.1.2.7.1

Move $e^{x}$ .

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

Step 7.3.1.2.7.2

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

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

Step 7.3.1.2.7.3

Add $x$ and $x$ .

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

Step 7.3.1.3

Simplify each term.

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Step 7.3.1.3.1

Multiply $- 1$ by $1$ .

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

Step 7.3.1.3.2

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

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Step 7.3.1.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 7.3.1.3.2.2

Multiply $x$ by $1$ .

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

Step 7.3.1.4

Expand $(- 1 + x e^{x}) (1 + e^{x})$ using the FOIL Method.

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Step 7.3.1.4.1

Apply the distributive property.

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

Step 7.3.1.4.2

Apply the distributive property.

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

Step 7.3.1.4.3

Apply the distributive property.

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

Step 7.3.1.5

Simplify each term.

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Step 7.3.1.5.1

Multiply $- 1$ by $1$ .

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

Step 7.3.1.5.2

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

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

Step 7.3.1.5.3

Multiply $x$ by $1$ .

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

Step 7.3.1.5.4

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

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Step 7.3.1.5.4.1

Move $e^{x}$ .

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

Step 7.3.1.5.4.2

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

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

Step 7.3.1.5.4.3

Add $x$ and $x$ .

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

Step 7.3.2

Combine the opposite terms in $- x^{2} e^{x} - x e^{x} - e^{2 x} x - e^{2 x} - 1 - e^{x} + x e^{x} + x e^{2 x}$ .

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Step 7.3.2.1

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

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

Step 7.3.2.2

Add $- x^{2} e^{x}$ and $0$ .

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

Step 7.3.2.3

Reorder the factors in the terms $- e^{2 x} x$ and $x e^{2 x}$ .

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

Step 7.3.2.4

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

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

Step 7.3.2.5

Add $- x^{2} e^{x}$ and $0$ .

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

Step 7.4

Reorder terms.

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

Step 7.5

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

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

Step 7.6

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

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

Step 7.7

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

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

Step 7.8

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

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

Step 7.9

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

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

Step 7.10

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

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

Step 7.11

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

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

Step 7.12

Rewrite $- (e^{x} x^{2} + 1 + e^{2 x} + e^{x})$ as $- 1 (e^{x} x^{2} + 1 + e^{2 x} + e^{x})$ .

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

Step 7.13

Move the negative in front of the fraction.

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

Step 7.14

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

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