Calculus Examples

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

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{4} e^{x}$ with respect to $x$ is $2 \frac{d}{d x} [x^{4} e^{x}]$ .

$2 \frac{d}{d x} [x^{4} e^{x}]$

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 $f (x) = x^{4}$ and $g (x) = e^{x}$ .

$2 (x^{4} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{4}])$

Step 3

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

$2 (x^{4} e^{x} + e^{x} \frac{d}{d x} [x^{4}])$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$2 (x^{4} e^{x} + e^{x} (4 x^{3}))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$2 (x^{4} e^{x}) + 2 (e^{x} (4 x^{3}))$

Step 5.2

Multiply $4$ by $2$ .

$2 x^{4} e^{x} + 8 e^{x} x^{3}$

Step 5.3

Reorder terms.

$2 e^{x} x^{4} + 8 e^{x} x^{3}$

Step 5.4

Reorder factors in $2 e^{x} x^{4} + 8 e^{x} x^{3}$ .

$2 x^{4} e^{x} + 8 x^{3} e^{x}$