Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

Step 3.2

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 3.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 3.3

Differentiate.

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Simplify the expression.

Tap for more steps...

Step 3.3.3.1

Multiply $2$ by $1$ .

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

Step 3.3.3.2

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

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

Step 3.3.4

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

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

Step 3.4

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

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

Step 3.5

Differentiate using the Constant Rule.

Tap for more steps...

Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.6

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

Tap for more steps...

Step 3.6.1

Move $e^{x}$ .

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

Step 3.6.2

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

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

Step 3.6.3

Add $x$ and $2 x$ .

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

Step 3.7

Simplify.

Tap for more steps...

Step 3.7.1

Apply the distributive property.

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

Step 3.7.2

Apply the distributive property.

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

Step 3.7.3

Simplify the numerator.

Tap for more steps...

Step 3.7.3.1

Simplify each term.

Tap for more steps...

Step 3.7.3.1.1

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

Tap for more steps...

Step 3.7.3.1.1.1

Move $e^{2 x}$ .

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

Step 3.7.3.1.1.2

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

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

Step 3.7.3.1.1.3

Add $2 x$ and $x$ .

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

Step 3.7.3.1.2

Multiply $2$ by $1$ .

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

Step 3.7.3.2

Subtract $e^{3 x}$ from $2 e^{3 x}$ .

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

Step 4

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

$y' = \frac{2 e^{2 x} + e^{3 x}}{{(e^{x} + 1)}^{2}}$

Step 5

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

$\frac{d y}{d x} = \frac{2 e^{2 x} + e^{3 x}}{{(e^{x} + 1)}^{2}}$