Calculus Examples

$f (x) = (x^{3} + 2 x) e^{x}$

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 $f (x) = x^{3} + 2 x$ and $g (x) = e^{x}$ .

$(x^{3} + 2 x) \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{3} + 2 x]$

Step 2

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

$(x^{3} + 2 x) e^{x} + e^{x} \frac{d}{d x} [x^{3} + 2 x]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{3} + 2 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [2 x]$ .

$(x^{3} + 2 x) e^{x} + e^{x} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [2 x])$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$(x^{3} + 2 x) e^{x} + e^{x} (3 x^{2} + \frac{d}{d x} [2 x])$

Step 3.3

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

$(x^{3} + 2 x) e^{x} + e^{x} (3 x^{2} + 2 \frac{d}{d x} [x])$

Step 3.4

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

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

Step 3.5

Multiply $2$ by $1$ .

$(x^{3} + 2 x) e^{x} + e^{x} (3 x^{2} + 2)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$x^{3} e^{x} + 2 x e^{x} + e^{x} (3 x^{2} + 2)$

Step 4.2

Apply the distributive property.

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

Step 4.3

Move $2$ to the left of $e^{x}$ .

$x^{3} e^{x} + 2 x e^{x} + e^{x} (3 x^{2}) + 2 e^{x}$

Step 4.4

Reorder terms.

$e^{x} x^{3} + 2 e^{x} x + 3 e^{x} x^{2} + 2 e^{x}$