Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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

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

Step 3.2

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

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Multiply $- 1$ by $1$ .

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

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Simplify the numerator.

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Step 3.4.2.1

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

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

Step 3.4.2.2

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

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Step 3.4.2.2.1

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

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

Step 3.4.2.2.2

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

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

Step 3.4.3

Reorder terms.

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

Step 3.4.4

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

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

Step 4

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

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

Step 5

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

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