Calculus Examples

$y = \frac{8 e^{x}}{2 e^{x} + 1}$

Step 1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8 e^{x}}{2 e^{x} + 1}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{e^{x}}{2 e^{x} + 1}]$ .

$8 \frac{d}{d x} [\frac{e^{x}}{2 e^{x} + 1}]$

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 $f (x) = e^{x}$ and $g (x) = 2 e^{x} + 1$ .

$8 \frac{(2 e^{x} + 1) \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [2 e^{x} + 1]}{{(2 e^{x} + 1)}^{2}}$

Step 3

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

$8 \frac{(2 e^{x} + 1) e^{x} - e^{x} \frac{d}{d x} [2 e^{x} + 1]}{{(2 e^{x} + 1)}^{2}}$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $2 e^{x} + 1$ with respect to $x$ is $\frac{d}{d x} [2 e^{x}] + \frac{d}{d x} [1]$ .

$8 \frac{(2 e^{x} + 1) e^{x} - e^{x} (\frac{d}{d x} [2 e^{x}] + \frac{d}{d x} [1])}{{(2 e^{x} + 1)}^{2}}$

Step 4.2

Since $2$ is constant with respect to $x$ , the derivative of $2 e^{x}$ with respect to $x$ is $2 \frac{d}{d x} [e^{x}]$ .

$8 \frac{(2 e^{x} + 1) e^{x} - e^{x} (2 \frac{d}{d x} [e^{x}] + \frac{d}{d x} [1])}{{(2 e^{x} + 1)}^{2}}$

Step 5

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

$8 \frac{(2 e^{x} + 1) e^{x} - e^{x} (2 e^{x} + \frac{d}{d x} [1])}{{(2 e^{x} + 1)}^{2}}$

Step 6

Differentiate using the Constant Rule.

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Step 6.1

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

$8 \frac{(2 e^{x} + 1) e^{x} - e^{x} (2 e^{x} + 0)}{{(2 e^{x} + 1)}^{2}}$

Step 6.2

Simplify the expression.

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Step 6.2.1

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

$8 \frac{(2 e^{x} + 1) e^{x} - e^{x} (2 e^{x})}{{(2 e^{x} + 1)}^{2}}$

Step 6.2.2

Multiply $2$ by $- 1$ .

$8 \frac{(2 e^{x} + 1) e^{x} - 2 e^{x} e^{x}}{{(2 e^{x} + 1)}^{2}}$

Step 7

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

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Step 7.1

Move $e^{x}$ .

$8 \frac{(2 e^{x} + 1) e^{x} - 2 (e^{x} e^{x})}{{(2 e^{x} + 1)}^{2}}$

Step 7.2

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

$8 \frac{(2 e^{x} + 1) e^{x} - 2 e^{x + x}}{{(2 e^{x} + 1)}^{2}}$

Step 7.3

Add $x$ and $x$ .

$8 \frac{(2 e^{x} + 1) e^{x} - 2 e^{2 x}}{{(2 e^{x} + 1)}^{2}}$

Step 8

Combine $8$ and $\frac{(2 e^{x} + 1) e^{x} - 2 e^{2 x}}{{(2 e^{x} + 1)}^{2}}$ .

$\frac{8 ((2 e^{x} + 1) e^{x} - 2 e^{2 x})}{{(2 e^{x} + 1)}^{2}}$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$\frac{8 (2 e^{x} e^{x} + 1 e^{x} - 2 e^{2 x})}{{(2 e^{x} + 1)}^{2}}$

Step 9.2

Apply the distributive property.

$\frac{8 (2 e^{x} e^{x}) + 8 (1 e^{x}) + 8 (- 2 e^{2 x})}{{(2 e^{x} + 1)}^{2}}$

Step 9.3

Simplify the numerator.

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Step 9.3.1

Simplify each term.

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Step 9.3.1.1

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

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Step 9.3.1.1.1

Move $e^{x}$ .

$\frac{8 (2 (e^{x} e^{x})) + 8 (1 e^{x}) + 8 (- 2 e^{2 x})}{{(2 e^{x} + 1)}^{2}}$

Step 9.3.1.1.2

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

$\frac{8 (2 e^{x + x}) + 8 (1 e^{x}) + 8 (- 2 e^{2 x})}{{(2 e^{x} + 1)}^{2}}$

Step 9.3.1.1.3

Add $x$ and $x$ .

$\frac{8 (2 e^{2 x}) + 8 (1 e^{x}) + 8 (- 2 e^{2 x})}{{(2 e^{x} + 1)}^{2}}$

Step 9.3.1.2

Multiply $2$ by $8$ .

$\frac{16 e^{2 x} + 8 (1 e^{x}) + 8 (- 2 e^{2 x})}{{(2 e^{x} + 1)}^{2}}$

Step 9.3.1.3

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

$\frac{16 e^{2 x} + 8 e^{x} + 8 (- 2 e^{2 x})}{{(2 e^{x} + 1)}^{2}}$

Step 9.3.1.4

Multiply $- 2$ by $8$ .

$\frac{16 e^{2 x} + 8 e^{x} - 16 e^{2 x}}{{(2 e^{x} + 1)}^{2}}$

Step 9.3.2

Combine the opposite terms in $16 e^{2 x} + 8 e^{x} - 16 e^{2 x}$ .

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Step 9.3.2.1

Subtract $16 e^{2 x}$ from $16 e^{2 x}$ .

$\frac{8 e^{x} + 0}{{(2 e^{x} + 1)}^{2}}$

Step 9.3.2.2

Add $8 e^{x}$ and $0$ .

$\frac{8 e^{x}}{{(2 e^{x} + 1)}^{2}}$