Calculus Examples

$f (x) = \frac{3 x (1 - e^{x})}{1 + e^{x}}$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3 x (1 - e^{x})}{1 + e^{x}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{x (1 - e^{x})}{1 + e^{x}}]$ .

$3 \frac{d}{d x} [\frac{x (1 - e^{x})}{1 + e^{x}}]$

Step 2

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

$3 \frac{(1 + e^{x}) \frac{d}{d x} [x (1 - e^{x})] - x (1 - e^{x}) \frac{d}{d x} [1 + e^{x}]}{{(1 + e^{x})}^{2}}$

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

$3 \frac{(1 + e^{x}) (x \frac{d}{d x} [1 - e^{x}] + (1 - e^{x}) \frac{d}{d x} [x]) - x (1 - e^{x}) \frac{d}{d x} [1 + e^{x}]}{{(1 + e^{x})}^{2}}$

Step 4

Differentiate.

Tap for more steps...

Step 4.1

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

$3 \frac{(1 + e^{x}) (x (\frac{d}{d x} [1] + \frac{d}{d x} [- e^{x}]) + (1 - e^{x}) \frac{d}{d x} [x]) - x (1 - e^{x}) \frac{d}{d x} [1 + e^{x}]}{{(1 + e^{x})}^{2}}$

Step 4.2

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

$3 \frac{(1 + e^{x}) (x (0 + \frac{d}{d x} [- e^{x}]) + (1 - e^{x}) \frac{d}{d x} [x]) - x (1 - e^{x}) \frac{d}{d x} [1 + e^{x}]}{{(1 + e^{x})}^{2}}$

Step 4.3

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

$3 \frac{(1 + e^{x}) (x \frac{d}{d x} [- e^{x}] + (1 - e^{x}) \frac{d}{d x} [x]) - x (1 - e^{x}) \frac{d}{d x} [1 + e^{x}]}{{(1 + e^{x})}^{2}}$

Step 4.4

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

$3 \frac{(1 + e^{x}) (x (- \frac{d}{d x} [e^{x}]) + (1 - e^{x}) \frac{d}{d x} [x]) - x (1 - e^{x}) \frac{d}{d x} [1 + e^{x}]}{{(1 + e^{x})}^{2}}$

Step 5

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

$3 \frac{(1 + e^{x}) (x (- e^{x}) + (1 - e^{x}) \frac{d}{d x} [x]) - x (1 - e^{x}) \frac{d}{d x} [1 + e^{x}]}{{(1 + e^{x})}^{2}}$

Step 6

Differentiate.

Tap for more steps...

Step 6.1

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

$3 \frac{(1 + e^{x}) (x (- e^{x}) + (1 - e^{x}) \cdot 1) - x (1 - e^{x}) \frac{d}{d x} [1 + e^{x}]}{{(1 + e^{x})}^{2}}$

Step 6.2

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

$3 \frac{(1 + e^{x}) (x (- e^{x}) + 1 - e^{x}) - x (1 - e^{x}) \frac{d}{d x} [1 + e^{x}]}{{(1 + e^{x})}^{2}}$

Step 6.3

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

$3 \frac{(1 + e^{x}) (x (- e^{x}) + 1 - e^{x}) - x (1 - e^{x}) (\frac{d}{d x} [1] + \frac{d}{d x} [e^{x}])}{{(1 + e^{x})}^{2}}$

Step 6.4

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

$3 \frac{(1 + e^{x}) (x (- e^{x}) + 1 - e^{x}) - x (1 - e^{x}) (0 + \frac{d}{d x} [e^{x}])}{{(1 + e^{x})}^{2}}$

Step 6.5

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

$3 \frac{(1 + e^{x}) (x (- e^{x}) + 1 - e^{x}) - x (1 - e^{x}) \frac{d}{d x} [e^{x}]}{{(1 + e^{x})}^{2}}$

Step 7

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

$3 \frac{(1 + e^{x}) (x (- e^{x}) + 1 - e^{x}) - x (1 - e^{x}) e^{x}}{{(1 + e^{x})}^{2}}$

Step 8

Combine $3$ and $\frac{(1 + e^{x}) (x (- e^{x}) + 1 - e^{x}) - x (1 - e^{x}) e^{x}}{{(1 + e^{x})}^{2}}$ .

$\frac{3 ((1 + e^{x}) (x (- e^{x}) + 1 - e^{x}) - x (1 - e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Apply the distributive property.

$\frac{3 ((1 + e^{x}) (x (- e^{x}) + 1 - e^{x}) + (- x \cdot 1 - x (- e^{x})) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.2

Apply the distributive property.

$\frac{3 ((1 + e^{x}) (x (- e^{x}) + 1 - e^{x}) - x \cdot 1 e^{x} - x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.3

Apply the distributive property.

$\frac{3 ((1 + e^{x}) (x (- e^{x}) + 1 - e^{x})) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4

Simplify the numerator.

Tap for more steps...

Step 9.4.1

Simplify each term.

Tap for more steps...

Step 9.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{3 ((1 + e^{x}) (- x e^{x} + 1 - e^{x})) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.2

Expand $(1 + e^{x}) (- x e^{x} + 1 - e^{x})$ by multiplying each term in the first expression by each term in the second expression.

$\frac{3 (1 (- x e^{x}) + 1 \cdot 1 + 1 (- e^{x}) + e^{x} (- x e^{x}) + e^{x} \cdot 1 + e^{x} (- e^{x})) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.3

Simplify each term.

Tap for more steps...

Step 9.4.1.3.1

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

$\frac{3 (- x e^{x} + 1 \cdot 1 + 1 (- e^{x}) + e^{x} (- x e^{x}) + e^{x} \cdot 1 + e^{x} (- e^{x})) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.3.2

Multiply $1$ by $1$ .

$\frac{3 (- x e^{x} + 1 + 1 (- e^{x}) + e^{x} (- x e^{x}) + e^{x} \cdot 1 + e^{x} (- e^{x})) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.3.3

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

$\frac{3 (- x e^{x} + 1 - e^{x} + e^{x} (- x e^{x}) + e^{x} \cdot 1 + e^{x} (- e^{x})) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.3.4

Rewrite using the commutative property of multiplication.

$\frac{3 (- x e^{x} + 1 - e^{x} - e^{x} (x e^{x}) + e^{x} \cdot 1 + e^{x} (- e^{x})) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.3.5

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

Tap for more steps...

Step 9.4.1.3.5.1

Move $e^{x}$ .

$\frac{3 (- x e^{x} + 1 - e^{x} - (e^{x} e^{x}) x + e^{x} \cdot 1 + e^{x} (- e^{x})) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.3.5.2

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

$\frac{3 (- x e^{x} + 1 - e^{x} - e^{x + x} x + e^{x} \cdot 1 + e^{x} (- e^{x})) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.3.5.3

Add $x$ and $x$ .

$\frac{3 (- x e^{x} + 1 - e^{x} - e^{2 x} x + e^{x} \cdot 1 + e^{x} (- e^{x})) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.3.6

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

$\frac{3 (- x e^{x} + 1 - e^{x} - e^{2 x} x + e^{x} + e^{x} (- e^{x})) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.3.7

Rewrite using the commutative property of multiplication.

$\frac{3 (- x e^{x} + 1 - e^{x} - e^{2 x} x + e^{x} - e^{x} e^{x}) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.3.8

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

Tap for more steps...

Step 9.4.1.3.8.1

Move $e^{x}$ .

$\frac{3 (- x e^{x} + 1 - e^{x} - e^{2 x} x + e^{x} - (e^{x} e^{x})) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.3.8.2

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

$\frac{3 (- x e^{x} + 1 - e^{x} - e^{2 x} x + e^{x} - e^{x + x}) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.3.8.3

Add $x$ and $x$ .

$\frac{3 (- x e^{x} + 1 - e^{x} - e^{2 x} x + e^{x} - e^{2 x}) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.4

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

Tap for more steps...

Step 9.4.1.4.1

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

$\frac{3 (- x e^{x} + 1 - e^{2 x} x + 0 - e^{2 x}) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.4.2

Add $- x e^{x} + 1 - e^{2 x} x$ and $0$ .

$\frac{3 (- x e^{x} + 1 - e^{2 x} x - e^{2 x}) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.5

Apply the distributive property.

$\frac{3 (- x e^{x}) + 3 \cdot 1 + 3 (- e^{2 x} x) + 3 (- e^{2 x}) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.6

Simplify.

Tap for more steps...

Step 9.4.1.6.1

Multiply $- 1$ by $3$ .

$\frac{- 3 (x e^{x}) + 3 \cdot 1 + 3 (- e^{2 x} x) + 3 (- e^{2 x}) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.6.2

Multiply $3$ by $1$ .

$\frac{- 3 (x e^{x}) + 3 + 3 (- e^{2 x} x) + 3 (- e^{2 x}) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.6.3

Multiply $- 1$ by $3$ .

$\frac{- 3 (x e^{x}) + 3 - 3 (e^{2 x} x) + 3 (- e^{2 x}) + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.6.4

Multiply $- 1$ by $3$ .

$\frac{- 3 (x e^{x}) + 3 - 3 (e^{2 x} x) - 3 e^{2 x} + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.7

Remove parentheses.

$\frac{- 3 x e^{x} + 3 - 3 e^{2 x} x - 3 e^{2 x} + 3 (- x \cdot 1 e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.8

Multiply $- 1$ by $1$ .

$\frac{- 3 x e^{x} + 3 - 3 e^{2 x} x - 3 e^{2 x} + 3 (- x e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.9

Multiply $- 1$ by $3$ .

$\frac{- 3 x e^{x} + 3 - 3 e^{2 x} x - 3 e^{2 x} - 3 (x e^{x}) + 3 (- x (- e^{x}) e^{x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.10

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

Tap for more steps...

Step 9.4.1.10.1

Move $e^{x}$ .

$\frac{- 3 x e^{x} + 3 - 3 e^{2 x} x - 3 e^{2 x} - 3 x e^{x} + 3 (- x (- (e^{x} e^{x})))}{{(1 + e^{x})}^{2}}$

Step 9.4.1.10.2

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

$\frac{- 3 x e^{x} + 3 - 3 e^{2 x} x - 3 e^{2 x} - 3 x e^{x} + 3 (- x (- e^{x + x}))}{{(1 + e^{x})}^{2}}$

Step 9.4.1.10.3

Add $x$ and $x$ .

$\frac{- 3 x e^{x} + 3 - 3 e^{2 x} x - 3 e^{2 x} - 3 x e^{x} + 3 (- x (- e^{2 x}))}{{(1 + e^{x})}^{2}}$

Step 9.4.1.11

Rewrite using the commutative property of multiplication.

$\frac{- 3 x e^{x} + 3 - 3 e^{2 x} x - 3 e^{2 x} - 3 x e^{x} + 3 (- 1 \cdot - 1 x e^{2 x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.12

Multiply $- 1$ by $- 1$ .

$\frac{- 3 x e^{x} + 3 - 3 e^{2 x} x - 3 e^{2 x} - 3 x e^{x} + 3 (1 x e^{2 x})}{{(1 + e^{x})}^{2}}$

Step 9.4.1.13

Multiply $x$ by $1$ .

$\frac{- 3 x e^{x} + 3 - 3 e^{2 x} x - 3 e^{2 x} - 3 x e^{x} + 3 x e^{2 x}}{{(1 + e^{x})}^{2}}$

Step 9.4.2

Combine the opposite terms in $- 3 x e^{x} + 3 - 3 e^{2 x} x - 3 e^{2 x} - 3 x e^{x} + 3 x e^{2 x}$ .

Tap for more steps...

Step 9.4.2.1

Reorder the factors in the terms $- 3 e^{2 x} x$ and $3 x e^{2 x}$ .

$\frac{- 3 x e^{x} + 3 - 3 e^{2 x} x - 3 e^{2 x} - 3 x e^{x} + 3 e^{2 x} x}{{(1 + e^{x})}^{2}}$

Step 9.4.2.2

Add $- 3 e^{2 x} x$ and $3 e^{2 x} x$ .

$\frac{- 3 x e^{x} + 3 + 0 - 3 e^{2 x} - 3 x e^{x}}{{(1 + e^{x})}^{2}}$

Step 9.4.2.3

Add $- 3 x e^{x} + 3$ and $0$ .

$\frac{- 3 x e^{x} + 3 - 3 e^{2 x} - 3 x e^{x}}{{(1 + e^{x})}^{2}}$

Step 9.4.3

Subtract $3 x e^{x}$ from $- 3 x e^{x}$ .

$\frac{- 6 x e^{x} + 3 - 3 e^{2 x}}{{(1 + e^{x})}^{2}}$

Step 9.5

Reorder terms.

$\frac{- 6 e^{x} x + 3 - 3 e^{2 x}}{{(1 + e^{x})}^{2}}$

Step 9.6

Factor $3$ out of $- 6 e^{x} x + 3 - 3 e^{2 x}$ .

Tap for more steps...

Step 9.6.1

Factor $3$ out of $- 6 e^{x} x$ .

$\frac{3 (- 2 e^{x} x) + 3 - 3 e^{2 x}}{{(1 + e^{x})}^{2}}$

Step 9.6.2

Factor $3$ out of $3$ .

$\frac{3 (- 2 e^{x} x) + 3 (1) - 3 e^{2 x}}{{(1 + e^{x})}^{2}}$

Step 9.6.3

Factor $3$ out of $- 3 e^{2 x}$ .

$\frac{3 (- 2 e^{x} x) + 3 (1) + 3 (- e^{2 x})}{{(1 + e^{x})}^{2}}$

Step 9.6.4

Factor $3$ out of $3 (- 2 e^{x} x) + 3 (1)$ .

$\frac{3 (- 2 e^{x} x + 1) + 3 (- e^{2 x})}{{(1 + e^{x})}^{2}}$

Step 9.6.5

Factor $3$ out of $3 (- 2 e^{x} x + 1) + 3 (- e^{2 x})$ .

$\frac{3 (- 2 e^{x} x + 1 - e^{2 x})}{{(1 + e^{x})}^{2}}$

Step 9.7

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

$\frac{3 (- (2 e^{x} x) + 1 - e^{2 x})}{{(1 + e^{x})}^{2}}$

Step 9.8

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

$\frac{3 (- (2 e^{x} x) - 1 (- 1) - e^{2 x})}{{(1 + e^{x})}^{2}}$

Step 9.9

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

$\frac{3 (- (2 e^{x} x - 1) - e^{2 x})}{{(1 + e^{x})}^{2}}$

Step 9.10

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

$\frac{3 (- (2 e^{x} x - 1) - (e^{2 x}))}{{(1 + e^{x})}^{2}}$

Step 9.11

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

$\frac{3 (- (2 e^{x} x - 1 + e^{2 x}))}{{(1 + e^{x})}^{2}}$

Step 9.12

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

$\frac{3 (- 1 (2 e^{x} x - 1 + e^{2 x}))}{{(1 + e^{x})}^{2}}$

Step 9.13

Move the negative in front of the fraction.

$- \frac{3 (2 e^{x} x - 1 + e^{2 x})}{{(1 + e^{x})}^{2}}$

Step 9.14

Reorder factors in $- \frac{3 (2 e^{x} x - 1 + e^{2 x})}{{(1 + e^{x})}^{2}}$ .

$- \frac{3 (2 x e^{x} - 1 + e^{2 x})}{{(1 + e^{x})}^{2}}$