Calculus Examples

$h (x) = x^{\sqrt{x}}$ ; $a = 1$

Step 1

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

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

Step 2

Use the properties of logarithms to simplify the differentiation.

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Step 2.1

Rewrite $x^{x^{\frac{1}{2}}}$ as $e^{\ln (x^{x^{\frac{1}{2}}})}$ .

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

Step 2.2

Expand $\ln (x^{x^{\frac{1}{2}}})$ by moving $x^{\frac{1}{2}}$ outside the logarithm.

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

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

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Step 3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}} \ln (x)$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}} \ln (x)$ .

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

Step 4

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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = \ln (x)$ .

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

Step 5

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

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

Step 6

Combine fractions.

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Step 6.1

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

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

Step 6.2

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 7

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 7.1

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

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Step 7.1.1

Raise $x$ to the power of $1$ .

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

Step 7.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.2

Write $1$ as a fraction with a common denominator.

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

Step 7.3

Combine the numerators over the common denominator.

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

Step 7.4

Subtract $1$ from $2$ .

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

Step 8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 10

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

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

Simplify the numerator.

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Step 12.1

Multiply $- 1$ by $2$ .

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

Step 12.2

Subtract $2$ from $1$ .

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

Step 13

Move the negative in front of the fraction.

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

Step 14

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

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

Step 15

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

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

Step 16

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 17

Simplify.

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Step 17.1

Apply the distributive property.

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

Step 17.2

Combine terms.

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Step 17.2.1

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

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

Step 17.2.2

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

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