Calculus Examples

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

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Combine fractions.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.4

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

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Step 4.4.1

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

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Step 4.4.1.1

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

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

Step 4.4.1.2

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

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

Step 4.4.2

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

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

Step 4.4.3

Combine the numerators over the common denominator.

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

Step 4.4.4

Subtract $1$ from $2$ .

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

Combine the numerators over the common denominator.

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

Step 4.9

Simplify the numerator.

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Step 4.9.1

Multiply $- 1$ by $2$ .

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

Step 4.9.2

Subtract $2$ from $1$ .

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

Step 4.10

Move the negative in front of the fraction.

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

Step 4.11

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

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

Step 4.12

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

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

Step 4.13

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

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

Step 5

Reform the equation by setting the left side equal to the right side.

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

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

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