Calculus Examples

$y = \ln (\frac{e^{2 x}}{1 + e^{2 x}})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (\frac{e^{2 x}}{1 + e^{2 x}}))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $g (x) = \frac{e^{2 x}}{1 + e^{2 x}}$ .

Tap for more steps...

Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{e^{2 x}}{1 + e^{2 x}}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u_{1}$ with $\frac{e^{2 x}}{1 + e^{2 x}}$ .

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

Step 3.2

Multiply by the reciprocal of the fraction to divide by $\frac{e^{2 x}}{1 + e^{2 x}}$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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 $g (x) = 2 x$ .

Tap for more steps...

Step 3.5.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

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

Step 3.5.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

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

Step 3.5.3

Replace all occurrences of $u_{2}$ with $2 x$ .

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

Step 3.6

Differentiate.

Tap for more steps...

Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

Simplify the expression.

Tap for more steps...

Step 3.6.3.1

Multiply $2$ by $1$ .

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

Step 3.6.3.2

Move $2$ to the left of $(1 + e^{2 x}) e^{2 x}$ .

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

Step 3.6.4

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

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

Step 3.6.5

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

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

Step 3.6.6

Add $0$ and $\frac{d}{d x} [e^{2 x}]$ .

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

Step 3.7

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 $g (x) = 2 x$ .

Tap for more steps...

Step 3.7.1

To apply the Chain Rule, set $u_{3}$ as $2 x$ .

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

Step 3.7.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $e$ .

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

Step 3.7.3

Replace all occurrences of $u_{3}$ with $2 x$ .

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

Step 3.8

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

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

Step 3.9

Add $2 x$ and $2 x$ .

$\frac{1 + e^{2 x}}{e^{2 x}} \cdot \frac{2 (1 + e^{2 x}) e^{2 x} - e^{4 x} \frac{d}{d x} [2 x]}{{(1 + e^{2 x})}^{2}}$

Step 3.10

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

$\frac{1 + e^{2 x}}{e^{2 x}} \cdot \frac{2 (1 + e^{2 x}) e^{2 x} - e^{4 x} (2 \frac{d}{d x} [x])}{{(1 + e^{2 x})}^{2}}$

Step 3.11

Multiply $2$ by $- 1$ .

$\frac{1 + e^{2 x}}{e^{2 x}} \cdot \frac{2 (1 + e^{2 x}) e^{2 x} - 2 e^{4 x} \frac{d}{d x} [x]}{{(1 + e^{2 x})}^{2}}$

Step 3.12

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

$\frac{1 + e^{2 x}}{e^{2 x}} \cdot \frac{2 (1 + e^{2 x}) e^{2 x} - 2 e^{4 x} \cdot 1}{{(1 + e^{2 x})}^{2}}$

Step 3.13

Combine fractions.

Tap for more steps...

Step 3.13.1

Multiply $- 2$ by $1$ .

$\frac{1 + e^{2 x}}{e^{2 x}} \cdot \frac{2 (1 + e^{2 x}) e^{2 x} - 2 e^{4 x}}{{(1 + e^{2 x})}^{2}}$

Step 3.13.2

Multiply $\frac{1 + e^{2 x}}{e^{2 x}}$ by $\frac{2 (1 + e^{2 x}) e^{2 x} - 2 e^{4 x}}{{(1 + e^{2 x})}^{2}}$ .

$\frac{(1 + e^{2 x}) (2 (1 + e^{2 x}) e^{2 x} - 2 e^{4 x})}{e^{2 x} {(1 + e^{2 x})}^{2}}$

Step 3.14

Cancel the common factors.

Tap for more steps...

Step 3.14.1

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

$\frac{(1 + e^{2 x}) (2 (1 + e^{2 x}) e^{2 x} - 2 e^{4 x})}{(1 + e^{2 x}) (e^{2 x} (1 + e^{2 x}))}$

Step 3.14.2

Cancel the common factor.

$\frac{(1 + e^{2 x}) (2 (1 + e^{2 x}) e^{2 x} - 2 e^{4 x})}{(1 + e^{2 x}) (e^{2 x} (1 + e^{2 x}))}$

Step 3.14.3

Rewrite the expression.

$\frac{2 (1 + e^{2 x}) e^{2 x} - 2 e^{4 x}}{e^{2 x} (1 + e^{2 x})}$

Step 3.15

Simplify.

Tap for more steps...

Step 3.15.1

Apply the distributive property.

$\frac{(2 \cdot 1 + 2 e^{2 x}) e^{2 x} - 2 e^{4 x}}{e^{2 x} (1 + e^{2 x})}$

Step 3.15.2

Apply the distributive property.

$\frac{2 \cdot 1 e^{2 x} + 2 e^{2 x} e^{2 x} - 2 e^{4 x}}{e^{2 x} (1 + e^{2 x})}$

Step 3.15.3

Apply the distributive property.

$\frac{2 \cdot 1 e^{2 x} + 2 e^{2 x} e^{2 x} - 2 e^{4 x}}{e^{2 x} \cdot 1 + e^{2 x} e^{2 x}}$

Step 3.15.4

Simplify the numerator.

Tap for more steps...

Step 3.15.4.1

Simplify each term.

Tap for more steps...

Step 3.15.4.1.1

Multiply $2$ by $1$ .

$\frac{2 e^{2 x} + 2 e^{2 x} e^{2 x} - 2 e^{4 x}}{e^{2 x} \cdot 1 + e^{2 x} e^{2 x}}$

Step 3.15.4.1.2

Multiply $e^{2 x}$ by $e^{2 x}$ by adding the exponents.

Tap for more steps...

Step 3.15.4.1.2.1

Move $e^{2 x}$ .

$\frac{2 e^{2 x} + 2 (e^{2 x} e^{2 x}) - 2 e^{4 x}}{e^{2 x} \cdot 1 + e^{2 x} e^{2 x}}$

Step 3.15.4.1.2.2

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

$\frac{2 e^{2 x} + 2 e^{2 x + 2 x} - 2 e^{4 x}}{e^{2 x} \cdot 1 + e^{2 x} e^{2 x}}$

Step 3.15.4.1.2.3

Add $2 x$ and $2 x$ .

$\frac{2 e^{2 x} + 2 e^{4 x} - 2 e^{4 x}}{e^{2 x} \cdot 1 + e^{2 x} e^{2 x}}$

Step 3.15.4.2

Combine the opposite terms in $2 e^{2 x} + 2 e^{4 x} - 2 e^{4 x}$ .

Tap for more steps...

Step 3.15.4.2.1

Subtract $2 e^{4 x}$ from $2 e^{4 x}$ .

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

Step 3.15.4.2.2

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

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

Step 3.15.5

Combine terms.

Tap for more steps...

Step 3.15.5.1

Multiply $e^{2 x}$ by $1$ .

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

Step 3.15.5.2

Multiply $e^{2 x}$ by $e^{2 x}$ by adding the exponents.

Tap for more steps...

Step 3.15.5.2.1

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

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

Step 3.15.5.2.2

Add $2 x$ and $2 x$ .

$\frac{2 e^{2 x}}{e^{2 x} + e^{4 x}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{2 e^{2 x}}{e^{2 x} + e^{4 x}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{2 e^{2 x}}{e^{2 x} + e^{4 x}}$