Calculus Examples

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

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

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

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

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

Differentiate using the Power Rule.

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

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

Simplify the expression.

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Remove parentheses around $2 x$ .

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

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

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

Multiply $2$ by $e^{x}$ .

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

Reorder terms.

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

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