Calculus Examples

$e^{\frac{1}{2} x^{2}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = \frac{1}{2} x^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{1}{2} x^{2}$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{1}{2} x^{2}]$

Step 1.2

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

$e^{u} \frac{d}{d x} [\frac{1}{2} x^{2}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{1}{2} x^{2}$ .

$e^{\frac{1}{2} x^{2}} \frac{d}{d x} [\frac{1}{2} x^{2}]$

Step 2

Differentiate.

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Step 2.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$e^{\frac{x^{2}}{2}} \frac{d}{d x} [\frac{1}{2} x^{2}]$

Step 2.2

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Step 2.3

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{2}]$ .

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

Step 2.4

Combine $\frac{1}{2}$ and $e^{\frac{x^{2}}{2}}$ .

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

Step 2.5

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

$\frac{e^{\frac{x^{2}}{2}}}{2} (2 x)$

Step 2.6

Simplify terms.

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Step 2.6.1

Combine $2$ and $\frac{e^{\frac{x^{2}}{2}}}{2}$ .

$\frac{2 e^{\frac{x^{2}}{2}}}{2} x$

Step 2.6.2

Combine $\frac{2 e^{\frac{x^{2}}{2}}}{2}$ and $x$ .

$\frac{2 e^{\frac{x^{2}}{2}} x}{2}$

Step 2.6.3

Cancel the common factor of $2$ .

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Step 2.6.3.1

Cancel the common factor.

$\frac{2 e^{\frac{x^{2}}{2}} x}{2}$

Step 2.6.3.2

Divide $e^{\frac{x^{2}}{2}} x$ by $1$ .

$e^{\frac{x^{2}}{2}} x$

Step 2.6.4

Reorder factors in $e^{\frac{x^{2}}{2}} x$ .

$x e^{\frac{x^{2}}{2}}$