Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

Combine fractions.

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Step 1.1.4.1

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

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

Step 1.1.4.2

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

$f' (x) = \frac{1}{x \cdot x^{- \frac{1}{2}}} + \ln (x) \frac{d}{d x} (x^{\frac{1}{2}})$

Step 1.1.5

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

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Step 1.1.5.1

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

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Step 1.1.5.1.1

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

$f' (x) = \frac{1}{x \cdot x^{- \frac{1}{2}}} + \ln (x) \frac{d}{d x} (x^{\frac{1}{2}})$

Step 1.1.5.1.2

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

$f' (x) = \frac{1}{x^{1 - \frac{1}{2}}} + \ln (x) \frac{d}{d x} (x^{\frac{1}{2}})$

Step 1.1.5.2

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

$f' (x) = \frac{1}{x^{\frac{2}{2} - \frac{1}{2}}} + \ln (x) \frac{d}{d x} (x^{\frac{1}{2}})$

Step 1.1.5.3

Combine the numerators over the common denominator.

$f' (x) = \frac{1}{x^{\frac{2 - 1}{2}}} + \ln (x) \frac{d}{d x} (x^{\frac{1}{2}})$

Step 1.1.5.4

Subtract $1$ from $2$ .

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

Step 1.1.6

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

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

Step 1.1.7

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

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

Step 1.1.8

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

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

Step 1.1.9

Combine the numerators over the common denominator.

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

Step 1.1.10

Simplify the numerator.

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Step 1.1.10.1

Multiply $- 1$ by $2$ .

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

Step 1.1.10.2

Subtract $2$ from $1$ .

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

Step 1.1.11

Move the negative in front of the fraction.

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

Step 1.1.12

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

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

Step 1.1.13

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

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

Step 1.1.14

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

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{1}{x^{\frac{1}{2}}} + \frac{\ln (x)}{2 x^{\frac{1}{2}}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{1}{x^{\frac{1}{2}}} + \frac{\ln (x)}{2 x^{\frac{1}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{1}{x^{\frac{1}{2}}} + \frac{\ln (x)}{2 x^{\frac{1}{2}}} = 0$

Step 2.2

Subtract $\frac{1}{x^{\frac{1}{2}}}$ from both sides of the equation.

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

Step 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

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

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Step 2.4.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.4.1.1.2

Cancel the common factor of $2$ .

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Step 2.4.1.1.2.1

Cancel the common factor.

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

Step 2.4.1.1.2.2

Rewrite the expression.

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

Step 2.4.1.1.3

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 2.4.1.1.3.1

Cancel the common factor.

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

Step 2.4.1.1.3.2

Rewrite the expression.

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

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $- \frac{1}{x^{\frac{1}{2}}} (2 x^{\frac{1}{2}})$ .

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Step 2.4.2.1.1

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 2.4.2.1.1.1

Move the leading negative in $- \frac{1}{x^{\frac{1}{2}}}$ into the numerator.

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

Step 2.4.2.1.1.2

Factor $x^{\frac{1}{2}}$ out of $2 x^{\frac{1}{2}}$ .

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

Step 2.4.2.1.1.3

Cancel the common factor.

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

Step 2.4.2.1.1.4

Rewrite the expression.

$\ln (x) = - 1 \cdot 2$

Step 2.4.2.1.2

Multiply $- 1$ by $2$ .

$\ln (x) = - 2$

Step 2.5

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x)} = e^{- 2}$

Step 2.6

Rewrite $\ln (x) = - 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- 2} = x$

Step 2.7

Solve for $x$ .

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Step 2.7.1

Rewrite the equation as $x = e^{- 2}$ .

$x = e^{- 2}$

Step 2.7.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{e^{2}}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 3.1.2

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 3.1.3

Anything raised to $1$ is the base itself.

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

Step 3.1.4

Anything raised to $1$ is the base itself.

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

Step 3.2

Set the denominator in $\frac{1}{\sqrt{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = 0^{2}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

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

${(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{2}$

Step 3.3.2.2.1.2

Simplify.

$x = 0^{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 3.4

Set the denominator in $\frac{\ln (x)}{2 \sqrt{x}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{x} = 0$

Step 3.5

Solve for $x$ .

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Step 3.5.1

To remove the radical on the left side of the equation, square both sides of the equation.

${(2 \sqrt{x})}^{2} = 0^{2}$

Step 3.5.2

Simplify each side of the equation.

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Step 3.5.2.1

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

${(2 x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Simplify ${(2 x^{\frac{1}{2}})}^{2}$ .

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Step 3.5.2.2.1.1

Apply the product rule to $2 x^{\frac{1}{2}}$ .

$2^{2} {(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.5.2.2.1.2

Raise $2$ to the power of $2$ .

$4 {(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.5.2.2.1.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.5.2.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.5.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.5.2.2.1.3.2.1

Cancel the common factor.

$4 x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.5.2.2.1.3.2.2

Rewrite the expression.

$4 x^{1} = 0^{2}$

Step 3.5.2.2.1.4

Simplify.

$4 x = 0^{2}$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Raising $0$ to any positive power yields $0$ .

$4 x = 0$

Step 3.5.3

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 3.5.3.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $4$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 3.5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Divide $0$ by $4$ .

$x = 0$

Step 3.6

Set the argument in $\ln (x)$ less than or equal to $0$ to find where the expression is undefined.

$x \leq 0$

Step 3.7

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 3.8

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 4

Evaluate $\sqrt{x} \ln (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{1}{e^{2}}$ .

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Step 4.1.1

Substitute $\frac{1}{e^{2}}$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Remove parentheses.

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

Step 4.1.2.2

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

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

Step 4.1.2.3

Any root of $1$ is $1$ .

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

Step 4.1.2.4

Pull terms out from under the radical, assuming positive real numbers.

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

Step 4.1.2.5

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

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

Step 4.1.2.6

Expand $\ln (e^{- 2})$ by moving $- 2$ outside the logarithm.

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

Step 4.1.2.7

The natural logarithm of $e$ is $1$ .

$\frac{1}{e} (- 2 \cdot 1)$

Step 4.1.2.8

Multiply $- 2$ by $1$ .

$\frac{1}{e} \cdot - 2$

Step 4.1.2.9

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

$\frac{- 2}{e}$

Step 4.1.2.10

Move the negative in front of the fraction.

$- \frac{2}{e}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

$\sqrt{0} \ln (0)$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Remove parentheses.

$\sqrt{0} \ln (0)$

Step 4.2.2.2

The natural logarithm of zero is undefined.

Undefined

Step 4.3

List all of the points.

$(\frac{1}{e^{2}}, - \frac{2}{e})$

Step 5