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

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

Step 2.2

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

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

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

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

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

Multiply $2$ by $- 1$ .

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $- 2$ and $2$ .

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Step 2.6.4.1

Factor $2$ out of $- 2 e^{- \frac{x^{2}}{2}} x$ .

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

Step 2.6.4.2

Cancel the common factors.

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Step 2.6.4.2.1

Factor $2$ out of $2$ .

$\frac{2 (- e^{- \frac{x^{2}}{2}} x)}{2 (1)}$

Step 2.6.4.2.2

Cancel the common factor.

$\frac{2 (- e^{- \frac{x^{2}}{2}} x)}{2 \cdot 1}$

Step 2.6.4.2.3

Rewrite the expression.

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

Step 2.6.4.2.4

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

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

Step 2.6.5

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

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