Calculus Examples

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

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

Step 1

Let

y = f (x)

$y = f (x)$ , take the natural logarithm of both sides

ln (y) = ln (f (x))

$\ln (y) = \ln (f (x))$ .

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

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

Step 2

Expand the right hand side.

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Step 2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

x^{\frac{1}{2}}

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

ln (y) = ln ({(x^{\frac{1}{2}})}^{x})

$\ln (y) = \ln ({(x^{\frac{1}{2}})}^{x})$

Step 2.2

Expand

ln ({(x^{\frac{1}{2}})}^{x})

$\ln ({(x^{\frac{1}{2}})}^{x})$ by moving

x

$x$ outside the logarithm.

ln (y) = x ln (x^{\frac{1}{2}})

$\ln (y) = x \ln (x^{\frac{1}{2}})$

Step 2.3

Expand

ln (x^{\frac{1}{2}})

$\ln (x^{\frac{1}{2}})$ by moving

\frac{1}{2}

$\frac{1}{2}$ outside the logarithm.

ln (y) = x (\frac{1}{2} ln (x))

$\ln (y) = x (\frac{1}{2} \ln (x))$

Step 2.4

Combine

\frac{1}{2}

$\frac{1}{2}$ and

x

$x$ .

ln (y) = \frac{x}{2} ln (x)

$\ln (y) = \frac{x}{2} \ln (x)$

Step 2.5

Combine

\frac{x}{2}

$\frac{x}{2}$ and

ln (x)

$\ln (x)$ .

ln (y) = \frac{x ln (x)}{2}

$\ln (y) = \frac{x \ln (x)}{2}$

ln (y) = \frac{x ln (x)}{2}

$\ln (y) = \frac{x \ln (x)}{2}$

Step 3

Differentiate the expression using the chain rule, keeping in mind that

y

$y$ is a function of

x

$x$ .

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Step 3.1

Differentiate the left hand side

ln (y)

$\ln (y)$ using the chain rule.

\frac{y'}{y} = \frac{x ln (x)}{2}

$\frac{y'}{y} = \frac{x \ln (x)}{2}$

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate

\frac{x ln (x)}{2}

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

\frac{y'}{y} = \frac{d}{d x} [\frac{x ln (x)}{2}]

$\frac{y'}{y} = \frac{d}{d x} [\frac{x \ln (x)}{2}]$

Step 3.2.2

Since

\frac{1}{2}

$\frac{1}{2}$ is constant with respect to

x

$x$ , the derivative of

\frac{x ln (x)}{2}

$\frac{x \ln (x)}{2}$ with respect to

x

$x$ is

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

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

\frac{y'}{y} = \frac{1}{2} \frac{d}{d x} [x ln (x)]

$\frac{y'}{y} = \frac{1}{2} \frac{d}{d x} [x \ln (x)]$

Step 3.2.3

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = x

$f (x) = x$ and

g (x) = ln (x)

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

\frac{y'}{y} = \frac{1}{2} (x \frac{d}{d x} [ln (x)] + ln (x) \frac{d}{d x} [x])

$\frac{y'}{y} = \frac{1}{2} (x \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x])$

Step 3.2.4

The derivative of

ln (x)

$\ln (x)$ with respect to

x

$x$ is

\frac{1}{x}

$\frac{1}{x}$ .

\frac{y'}{y} = \frac{1}{2} (x \frac{1}{x} + ln (x) \frac{d}{d x} [x])

$\frac{y'}{y} = \frac{1}{2} (x \frac{1}{x} + \ln (x) \frac{d}{d x} [x])$

Step 3.2.5

Differentiate using the Power Rule.

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Step 3.2.5.1

Combine

x

$x$ and

\frac{1}{x}

$\frac{1}{x}$ .

\frac{y'}{y} = \frac{1}{2} (\frac{x}{x} + ln (x) \frac{d}{d x} [x])

$\frac{y'}{y} = \frac{1}{2} (\frac{x}{x} + \ln (x) \frac{d}{d x} [x])$

Step 3.2.5.2

Cancel the common factor of

x

$x$ .

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Step 3.2.5.2.1

Cancel the common factor.

$\frac{y'}{y} = \frac{1}{2} (\frac{x}{x} + \ln (x) \frac{d}{d x} [x])$

Step 3.2.5.2.2

Rewrite the expression.

$\frac{y'}{y} = \frac{1}{2} (1 + \ln (x) \frac{d}{d x} [x])$

Step 3.2.5.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$\frac{y'}{y} = \frac{1}{2} (1 + \ln (x) \cdot 1)$

Step 3.2.5.4

Multiply

$\ln (x)$ by

$1$ .

$\frac{y'}{y} = \frac{1}{2} (1 + \ln (x))$

Step 3.2.6

Simplify.

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Step 3.2.6.1

Apply the distributive property.

$\frac{y'}{y} = \frac{1}{2} \cdot 1 + \frac{1}{2} \ln (x)$

Step 3.2.6.2

Multiply

$\frac{1}{2}$ by

$1$ .

$\frac{y'}{y} = \frac{1}{2} + \frac{1}{2} \ln (x)$

Step 3.2.6.3

Reorder terms.

$\frac{y'}{y} = \frac{1}{2} \ln (x) + \frac{1}{2}$

Step 3.2.6.4

Combine

$\frac{1}{2}$ and

$\ln (x)$ .

$\frac{y'}{y} = \frac{\ln (x)}{2} + \frac{1}{2}$

Step 4

Isolate

$y'$ and substitute the original function for

$y$ in the right hand side.

$y' = (\frac{\ln (x)}{2} + \frac{1}{2}) {(\sqrt{x})}^{x}$

Step 5

Simplify the right hand side.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite

$\frac{\ln (x)}{2}$ as

$\frac{1}{2} \ln (x)$ .

$y' = (\frac{1}{2} \ln (x) + \frac{1}{2}) {(\sqrt{x})}^{x}$

Step 5.1.2

Simplify

$\frac{1}{2} \ln (x)$ by moving

$\frac{1}{2}$ inside the logarithm.

$y' = (\ln (x^{\frac{1}{2}}) + \frac{1}{2}) {(\sqrt{x})}^{x}$

Step 5.2

Apply the distributive property.

$y' = \ln (x^{\frac{1}{2}}) {\sqrt{x}}^{x} + \frac{1}{2} {\sqrt{x}}^{x}$

Step 5.3

Combine

$\frac{1}{2}$ and

${\sqrt{x}}^{x}$ .

$y' = \ln (x^{\frac{1}{2}}) {\sqrt{x}}^{x} + \frac{{\sqrt{x}}^{x}}{2}$

Step 5.4

Reorder factors in

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

$y' = {\sqrt{x}}^{x} \ln (x^{\frac{1}{2}}) + \frac{{\sqrt{x}}^{x}}{2}$