Calculus Examples

$2 x e^{x}$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Reorder terms.

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

Step 5.3

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

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