Calculus Examples

$y = {(\sqrt{x})}^{x}$

Step 1

Apply basic rules of exponents.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 1.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{x}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.2

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

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

Step 2

Use the properties of logarithms to simplify the differentiation.

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Step 2.1

Rewrite $x^{\frac{x}{2}}$ as $e^{\ln (x^{\frac{x}{2}})}$ .

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

Step 2.2

Expand $\ln (x^{\frac{x}{2}})$ by moving $\frac{x}{2}$ outside the logarithm.

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

Step 3

Combine $\frac{x}{2}$ and $\ln (x)$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Differentiate using the Constant Multiple Rule.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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 $g (x) = \ln (x)$ .

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

Step 7

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

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

Step 8

Differentiate using the Power Rule.

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Step 8.1

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

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

Step 8.2

Cancel the common factor of $x$ .

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Step 8.2.1

Cancel the common factor.

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

Step 8.2.2

Rewrite the expression.

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

Step 8.3

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

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

Step 8.4

Multiply $\ln (x)$ by $1$ .

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Combine terms.

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Step 9.2.1

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

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

Step 9.2.2

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

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