Calculus Examples

$y = \sqrt{e^{x}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{e^{x}}$ as $e^{\frac{x}{2}}$ .

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

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (e^{\frac{x}{2}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

Differentiate.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{e^{\frac{x}{2}}}{2}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{e^{\frac{x}{2}}}{2}$