Find the Derivative – d/dx y = square root of x natural log of x

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

Rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

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

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

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

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

Combine fractions.

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

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

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

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

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

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

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

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

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

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

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

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

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

Multiply $2$ by $1$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $1$ by $2$ .

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

Subtract $1$ from $2$ .

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

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

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

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

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

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

Multiply $2$ by $1$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

Find the Derivative – d/dx y = square root of x natural log of x

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