Calculus Examples

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

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

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$(x^{2} - 3) e^{x} + e^{x} (2 x + 0)$

Step 3.4

Add $2 x$ and $0$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Reorder terms.

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

Step 4.3

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

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