Calculus Examples

$y = 2 x \log (\sqrt{x})$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log (x)$ and $g (x) = x^{\frac{1}{2}}$ .

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Step 3.1

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

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

Step 3.2

The derivative of $\log (u)$ with respect to $u$ is $\frac{1}{u \ln (10)}$ .

$2 (x (\frac{1}{u \ln (10)} \frac{d}{d x} [x^{\frac{1}{2}}]) + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 3.3

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

$2 (x (\frac{1}{x^{\frac{1}{2}} \ln (10)} \frac{d}{d x} [x^{\frac{1}{2}}]) + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4

Combine fractions.

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Step 4.1

Combine $\frac{1}{x^{\frac{1}{2}} \ln (10)}$ and $x$ .

$2 (\frac{x}{x^{\frac{1}{2}} \ln (10)} \frac{d}{d x} [x^{\frac{1}{2}}] + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 4.2

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

$2 (\frac{x \cdot x^{- \frac{1}{2}}}{\ln (10)} \frac{d}{d x} [x^{\frac{1}{2}}] + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 5

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

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Step 5.1

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

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Step 5.1.1

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

$2 (\frac{x^{1} x^{- \frac{1}{2}}}{\ln (10)} \frac{d}{d x} [x^{\frac{1}{2}}] + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 5.1.2

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

$2 (\frac{x^{1 - \frac{1}{2}}}{\ln (10)} \frac{d}{d x} [x^{\frac{1}{2}}] + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 5.2

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

$2 (\frac{x^{\frac{2}{2} - \frac{1}{2}}}{\ln (10)} \frac{d}{d x} [x^{\frac{1}{2}}] + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 5.3

Combine the numerators over the common denominator.

$2 (\frac{x^{\frac{2 - 1}{2}}}{\ln (10)} \frac{d}{d x} [x^{\frac{1}{2}}] + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 5.4

Subtract $1$ from $2$ .

$2 (\frac{x^{\frac{1}{2}}}{\ln (10)} \frac{d}{d x} [x^{\frac{1}{2}}] + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 6

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

$2 (\frac{x^{\frac{1}{2}}}{\ln (10)} (\frac{1}{2} x^{\frac{1}{2} - 1}) + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 7

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

$2 (\frac{x^{\frac{1}{2}}}{\ln (10)} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 8

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

$2 (\frac{x^{\frac{1}{2}}}{\ln (10)} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 9

Combine the numerators over the common denominator.

$2 (\frac{x^{\frac{1}{2}}}{\ln (10)} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$2 (\frac{x^{\frac{1}{2}}}{\ln (10)} (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 10.2

Subtract $2$ from $1$ .

$2 (\frac{x^{\frac{1}{2}}}{\ln (10)} (\frac{1}{2} x^{\frac{- 1}{2}}) + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 11

Move the negative in front of the fraction.

$2 (\frac{x^{\frac{1}{2}}}{\ln (10)} (\frac{1}{2} x^{- \frac{1}{2}}) + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 12

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

$2 (\frac{x^{\frac{1}{2}}}{\ln (10)} \cdot \frac{x^{- \frac{1}{2}}}{2} + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 13

Multiply $\frac{x^{\frac{1}{2}}}{\ln (10)}$ by $\frac{x^{- \frac{1}{2}}}{2}$ .

$2 (\frac{x^{\frac{1}{2}} x^{- \frac{1}{2}}}{\ln (10) \cdot 2} + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 14

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

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Step 14.1

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

$2 (\frac{x^{\frac{1}{2} - \frac{1}{2}}}{\ln (10) \cdot 2} + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 14.2

Combine the numerators over the common denominator.

$2 (\frac{x^{\frac{1 - 1}{2}}}{\ln (10) \cdot 2} + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 14.3

Subtract $1$ from $1$ .

$2 (\frac{x^{\frac{0}{2}}}{\ln (10) \cdot 2} + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 14.4

Divide $0$ by $2$ .

$2 (\frac{x^{0}}{\ln (10) \cdot 2} + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 15

Simplify the expression.

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Step 15.1

Simplify $x^{0}$ .

$2 (\frac{1}{\ln (10) \cdot 2} + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 15.2

Move $2$ to the left of $\ln (10)$ .

$2 (\frac{1}{2 \cdot \ln (10)} + \log (x^{\frac{1}{2}}) \frac{d}{d x} [x])$

Step 16

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

$2 (\frac{1}{2 \ln (10)} + \log (x^{\frac{1}{2}}) \cdot 1)$

Step 17

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

$2 (\frac{1}{2 \ln (10)} + \log (x^{\frac{1}{2}}))$

Step 18

Simplify.

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Step 18.1

Apply the distributive property.

$2 \frac{1}{2 \ln (10)} + 2 \log (x^{\frac{1}{2}})$

Step 18.2

Combine terms.

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Step 18.2.1

Combine $2$ and $\frac{1}{2 \ln (10)}$ .

$\frac{2}{2 \ln (10)} + 2 \log (x^{\frac{1}{2}})$

Step 18.2.2

Cancel the common factor of $2$ .

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Step 18.2.2.1

Cancel the common factor.

$\frac{2}{2 \ln (10)} + 2 \log (x^{\frac{1}{2}})$

Step 18.2.2.2

Rewrite the expression.

$\frac{1}{\ln (10)} + 2 \log (x^{\frac{1}{2}})$

Step 18.3

Reorder terms.

$2 \log (x^{\frac{1}{2}}) + \frac{1}{\ln (10)}$