Calculus Examples

$f (x) = 32 x^{0.25}$

Step 1

Find the first derivative of the function.

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Step 1.1

Since $32$ is constant with respect to $x$ , the derivative of $32 x^{0.25}$ with respect to $x$ is $32 \frac{d}{d x} [x^{0.25}]$ .

$32 \frac{d}{d x} [x^{0.25}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 0.25$ .

$32 (0.25 x^{- 0.75})$

Step 1.3

Multiply $0.25$ by $32$ .

$8 x^{- 0.75}$

Step 1.4

Simplify.

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Step 1.4.1

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

$8 \frac{1}{x^{0.75}}$

Step 1.4.2

Combine $8$ and $\frac{1}{x^{0.75}}$ .

$\frac{8}{x^{0.75}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8}{x^{0.75}}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{1}{x^{0.75}}]$ .

$f'' (x) = 8 \frac{d}{d x} (\frac{1}{x^{0.75}})$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

Rewrite $\frac{1}{x^{0.75}}$ as ${(x^{0.75})}^{- 1}$ .

$f'' (x) = 8 \frac{d}{d x} ({(x^{0.75})}^{- 1})$

Step 2.2.2

Multiply the exponents in ${(x^{0.75})}^{- 1}$ .

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Step 2.2.2.1

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

$f'' (x) = 8 \frac{d}{d x} (x^{0.75 \cdot - 1})$

Step 2.2.2.2

Multiply $0.75$ by $- 1$ .

$f'' (x) = 8 \frac{d}{d x} (x^{- 0.75})$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 0.75$ .

$f'' (x) = 8 (- 0.75 x^{- 1.75})$

Step 2.4

Multiply $- 0.75$ by $8$ .

$f'' (x) = - 6 x^{- 1.75}$

Step 2.5

Simplify.

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Step 2.5.1

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

$f'' (x) = - 6 \frac{1}{x^{1.75}}$

Step 2.5.2

Combine terms.

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Step 2.5.2.1

Combine $- 6$ and $\frac{1}{x^{1.75}}$ .

$f'' (x) = \frac{- 6}{x^{1.75}}$

Step 2.5.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{6}{x^{1.75}}$

Step 3

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

$\frac{8}{x^{0.75}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Since $32$ is constant with respect to $x$ , the derivative of $32 x^{0.25}$ with respect to $x$ is $32 \frac{d}{d x} [x^{0.25}]$ .

$32 \frac{d}{d x} [x^{0.25}]$

Step 4.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 0.25$ .

$32 (0.25 x^{- 0.75})$

Step 4.1.3

Multiply $0.25$ by $32$ .

$8 x^{- 0.75}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

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

$8 \frac{1}{x^{0.75}}$

Step 4.1.4.2

Combine $8$ and $\frac{1}{x^{0.75}}$ .

$f' (x) = \frac{8}{x^{0.75}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{8}{x^{0.75}}$ .

$\frac{8}{x^{0.75}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{8}{x^{0.75}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{8}{x^{0.75}} = 0$

Step 5.2

Set the numerator equal to zero.

$8 = 0$

Step 5.3

Since $8 \neq 0$ , there are no solutions.

No solution

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

Change $0.75$ into a fraction.

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Step 6.1.1.1

Multiply by $\frac{100}{100}$ to remove the decimal.

$\frac{8}{x^{\frac{100 \cdot 0.75}{100}}}$

Step 6.1.1.2

Multiply $100$ by $0.75$ .

$\frac{8}{x^{\frac{75}{100}}}$

Step 6.1.1.3

Cancel the common factor of $75$ and $100$ .

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Step 6.1.1.3.1

Factor $25$ out of $75$ .

$\frac{8}{x^{\frac{25 (3)}{100}}}$

Step 6.1.1.3.2

Cancel the common factors.

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Step 6.1.1.3.2.1

Factor $25$ out of $100$ .

$\frac{8}{x^{\frac{25 \cdot 3}{25 \cdot 4}}}$

Step 6.1.1.3.2.2

Cancel the common factor.

$\frac{8}{x^{\frac{25 \cdot 3}{25 \cdot 4}}}$

Step 6.1.1.3.2.3

Rewrite the expression.

$\frac{8}{x^{\frac{3}{4}}}$

Step 6.1.2

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

$\frac{8}{\sqrt[4]{x^{3}}}$

Step 6.2

Set the denominator in $\frac{8}{\sqrt[4]{x^{3}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[4]{x^{3}} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{x^{3}}}^{4} = 0^{4}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

${(x^{\frac{3}{4}})}^{4} = 0^{4}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Multiply the exponents in ${(x^{\frac{3}{4}})}^{4}$ .

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Step 6.3.2.2.1.1

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

$x^{\frac{3}{4} \cdot 4} = 0^{4}$

Step 6.3.2.2.1.2

Cancel the common factor of $4$ .

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Step 6.3.2.2.1.2.1

Cancel the common factor.

$x^{\frac{3}{4} \cdot 4} = 0^{4}$

Step 6.3.2.2.1.2.2

Rewrite the expression.

$x^{3} = 0^{4}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

$x^{3} = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Step 6.3.3.2

Simplify $\sqrt[3]{0}$ .

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Step 6.3.3.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 6.3.3.2.2

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

$x = 0$

Step 6.4

Set the radicand in $\sqrt[4]{x^{3}}$ less than $0$ to find where the expression is undefined.

$x^{3} < 0$

Step 6.5

Solve for $x$ .

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Step 6.5.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt[3]{x^{3}} < \sqrt[3]{0}$

Step 6.5.2

Simplify the equation.

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Step 6.5.2.1

Simplify the left side.

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Step 6.5.2.1.1

Pull terms out from under the radical.

$x < \sqrt[3]{0}$

Step 6.5.2.2

Simplify the right side.

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Step 6.5.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 6.5.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x < \sqrt[3]{0^{3}}$

Step 6.5.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 6.6

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 7

Critical points to evaluate.

$x = 0$

Step 8

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.

$- \frac{6}{{(0)}^{1.75}}$

Step 9

Evaluate the second derivative.

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Step 9.1

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

$- \frac{6}{0}$

Step 9.2

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

Undefined

Step 10

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

No Local Extrema

Step 11