Calculus Examples

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

Step 1

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

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

Step 2

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 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Simplify terms.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor of $2$ .

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Step 3.2.3.1

Cancel the common factor.

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

Step 3.2.3.2

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

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

Step 3.3

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

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

Step 3.4

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

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