Algebra Examples

$\sqrt{x}$

Find the first derivative of the function.

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

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

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

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

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

$\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

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

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

Combine the numerators over the common denominator.

$\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}$

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Move the negative in front of the fraction.

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

Simplify.

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

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

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

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

Find the second derivative of the function.

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

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

Apply basic rules of exponents.

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

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

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

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

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

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

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $- 1$ .

$f'' (x) = \frac{1}{2} \cdot (- \frac{1}{2} x^{\frac{- 3}{2}})$

Move the negative in front of the fraction.

$f'' (x) = \frac{1}{2} \cdot (- \frac{1}{2} x^{- \frac{3}{2}})$

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

$f'' (x) = \frac{1}{2} \cdot (- \frac{x^{- \frac{3}{2}}}{2})$

Multiply $\frac{1}{2}$ by $\frac{x^{- \frac{3}{2}}}{2}$ .

$f'' (x) = - \frac{x^{- \frac{3}{2}}}{2 \cdot 2}$

Simplify the expression.

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Multiply $2$ by $2$ .

$f'' (x) = - \frac{x^{- \frac{3}{2}}}{4}$

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{1}{4 x^{\frac{3}{2}}}$

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

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

Find the first derivative.

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Find the first derivative.

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Move the negative in front of the fraction.

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

Simplify.

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

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

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

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

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

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

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

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Set the first derivative equal to $0$ .

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

Set the numerator equal to zero.

$1 = 0$

Since $1 \neq 0$ , there are no solutions.

No solution

Find the values where the derivative is undefined.

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Convert expressions with fractional exponents to radicals.

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

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

Anything raised to $1$ is the base itself.

$\frac{1}{2 \sqrt{x}}$

Set the denominator in $\frac{1}{2 \sqrt{x}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{x} = 0$

Solve for $x$ .

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$4 x = 0^{2}$

Simplify the right side.

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

$4 x = 0$

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

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

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

Simplify the left side.

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Cancel the common factor of $4$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

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Divide $0$ by $4$ .

$x = 0$

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

$x < 0$

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 $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Critical points to evaluate.

$x = 0$

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{1}{4 {(0)}^{\frac{3}{2}}}$

Evaluate the second derivative.

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Simplify the expression.

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Rewrite $0$ as $0^{2}$ .

$- \frac{1}{4 {(0^{2})}^{\frac{3}{2}}}$

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

$- \frac{1}{4 \cdot 0^{2 (\frac{3}{2})}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$- \frac{1}{4 \cdot 0^{2 (\frac{3}{2})}}$

Rewrite the expression.

$- \frac{1}{4 \cdot 0^{3}}$

Simplify the expression.

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

$- \frac{1}{4 \cdot 0}$

Multiply $4$ by $0$ .

$- \frac{1}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{1}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

Since the first derivative test failed, there are no local extrema.

No Local Extrema