Calculus Examples

$e^{x} (1 + x)$

Step 1

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

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

$e^{x} + (1 + x) e^{x}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$e^{x} + 1 e^{x} + x e^{x}$

Step 4.2

Combine terms.

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Step 4.2.1

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

$e^{x} + e^{x} + x e^{x}$

Step 4.2.2

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

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

Step 4.3

Reorder terms.

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

Step 4.4

Reorder factors in $e^{x} x + 2 e^{x}$ .

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