Calculus Examples

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

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Move the negative in front of the fraction.

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

Step 1.2

Combine fractions.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

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

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

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

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

Step 3

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 $g (x) = - \frac{x}{2}$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

Combine fractions.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

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

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

Step 4.4

Multiply $- 1$ by $1$ .

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

Step 4.5

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

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

Step 4.6

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Combine terms.

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $2$ .

$- \frac{x e^{- \frac{x}{2}}}{4} + \frac{1}{2} e^{- \frac{x}{2}}$

Step 5.2.3

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

$- \frac{x e^{- \frac{x}{2}}}{4} + \frac{e^{- \frac{x}{2}}}{2}$