Calculus Examples

$3 \sqrt{x} \ln (x)$

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

$\frac{d}{d x} [3 x^{\frac{1}{2}} \ln (x)]$

Step 1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{\frac{1}{2}} \ln (x)$ with respect to $x$ is $3 \frac{d}{d x} [x^{\frac{1}{2}} \ln (x)]$ .

$3 \frac{d}{d x} [x^{\frac{1}{2}} \ln (x)]$

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

$3 (x^{\frac{1}{2}} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 3

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

$3 (x^{\frac{1}{2}} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4

Combine fractions.

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Step 4.1

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

$3 (\frac{x^{\frac{1}{2}}}{x} + \ln (x) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4.2

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

$3 (\frac{1}{x \cdot x^{- \frac{1}{2}}} + \ln (x) \frac{d}{d x} [x^{\frac{1}{2}}])$

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 $1$ .

$3 (\frac{1}{x^{1} x^{- \frac{1}{2}}} + \ln (x) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 5.1.2

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

$3 (\frac{1}{x^{1 - \frac{1}{2}}} + \ln (x) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 5.2

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

$3 (\frac{1}{x^{\frac{2}{2} - \frac{1}{2}}} + \ln (x) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 5.3

Combine the numerators over the common denominator.

$3 (\frac{1}{x^{\frac{2 - 1}{2}}} + \ln (x) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 5.4

Subtract $1$ from $2$ .

$3 (\frac{1}{x^{\frac{1}{2}}} + \ln (x) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 6

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

$3 (\frac{1}{x^{\frac{1}{2}}} + \ln (x) (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 7

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

$3 (\frac{1}{x^{\frac{1}{2}}} + \ln (x) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 8

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

$3 (\frac{1}{x^{\frac{1}{2}}} + \ln (x) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 9

Combine the numerators over the common denominator.

$3 (\frac{1}{x^{\frac{1}{2}}} + \ln (x) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$3 (\frac{1}{x^{\frac{1}{2}}} + \ln (x) (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 10.2

Subtract $2$ from $1$ .

$3 (\frac{1}{x^{\frac{1}{2}}} + \ln (x) (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 11

Move the negative in front of the fraction.

$3 (\frac{1}{x^{\frac{1}{2}}} + \ln (x) (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 12

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

$3 (\frac{1}{x^{\frac{1}{2}}} + \ln (x) \frac{x^{- \frac{1}{2}}}{2})$

Step 13

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

$3 (\frac{1}{x^{\frac{1}{2}}} + \frac{\ln (x) x^{- \frac{1}{2}}}{2})$

Step 14

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

$3 (\frac{1}{x^{\frac{1}{2}}} + \frac{\ln (x)}{2 x^{\frac{1}{2}}})$

Step 15

Simplify.

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Step 15.1

Apply the distributive property.

$3 \frac{1}{x^{\frac{1}{2}}} + 3 \frac{\ln (x)}{2 x^{\frac{1}{2}}}$

Step 15.2

Combine terms.

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Step 15.2.1

Combine $3$ and $\frac{1}{x^{\frac{1}{2}}}$ .

$\frac{3}{x^{\frac{1}{2}}} + 3 \frac{\ln (x)}{2 x^{\frac{1}{2}}}$

Step 15.2.2

Combine $3$ and $\frac{\ln (x)}{2 x^{\frac{1}{2}}}$ .

$\frac{3}{x^{\frac{1}{2}}} + \frac{3 \ln (x)}{2 x^{\frac{1}{2}}}$