Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Add $2 x e^{x}$ and $e^{x} (2 x)$ .

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Step 4.2.1

Move $e^{x}$ .

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

Step 4.2.2

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

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

Step 4.3

Reorder terms.

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

Step 4.4

Reorder factors in $4 e^{x} x + e^{x} x^{2} + 2 e^{x}$ .

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