Calculus Examples

$C (x) = 375 + 0.75 x^{\frac{3}{4}}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $375 + 0.75 x^{\frac{3}{4}}$ with respect to $x$ is $\frac{d}{d x} [375] + \frac{d}{d x} [0.75 x^{\frac{3}{4}}]$ .

$\frac{d}{d x} [375] + \frac{d}{d x} [0.75 x^{\frac{3}{4}}]$

Step 1.1.2

Since $375$ is constant with respect to $x$ , the derivative of $375$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [0.75 x^{\frac{3}{4}}]$

Step 1.2

Evaluate $\frac{d}{d x} [0.75 x^{\frac{3}{4}}]$ .

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Step 1.2.1

Since $0.75$ is constant with respect to $x$ , the derivative of $0.75 x^{\frac{3}{4}}$ with respect to $x$ is $0.75 \frac{d}{d x} [x^{\frac{3}{4}}]$ .

$0 + 0.75 \frac{d}{d x} [x^{\frac{3}{4}}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{4}$ .

$0 + 0.75 (\frac{3}{4} x^{\frac{3}{4} - 1})$

Step 1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$0 + 0.75 (\frac{3}{4} x^{\frac{3}{4} - 1 \cdot \frac{4}{4}})$

Step 1.2.4

Combine $- 1$ and $\frac{4}{4}$ .

$0 + 0.75 (\frac{3}{4} x^{\frac{3}{4} + \frac{- 1 \cdot 4}{4}})$

Step 1.2.5

Combine the numerators over the common denominator.

$0 + 0.75 (\frac{3}{4} x^{\frac{3 - 1 \cdot 4}{4}})$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $4$ .

$0 + 0.75 (\frac{3}{4} x^{\frac{3 - 4}{4}})$

Step 1.2.6.2

Subtract $4$ from $3$ .

$0 + 0.75 (\frac{3}{4} x^{\frac{- 1}{4}})$

Step 1.2.7

Move the negative in front of the fraction.

$0 + 0.75 (\frac{3}{4} x^{- \frac{1}{4}})$

Step 1.2.8

Combine $\frac{3}{4}$ and $x^{- \frac{1}{4}}$ .

$0 + 0.75 \frac{3 x^{- \frac{1}{4}}}{4}$

Step 1.2.9

Combine $0.75$ and $\frac{3 x^{- \frac{1}{4}}}{4}$ .

$0 + \frac{0.75 (3 x^{- \frac{1}{4}})}{4}$

Step 1.2.10

Multiply $3$ by $0.75$ .

$0 + \frac{2.25 x^{- \frac{1}{4}}}{4}$

Step 1.2.11

Move $x^{- \frac{1}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 + \frac{2.25}{4 x^{\frac{1}{4}}}$

Step 1.3

Simplify.

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Step 1.3.1

Add $0$ and $\frac{2.25}{4 x^{\frac{1}{4}}}$ .

$\frac{2.25}{4 x^{\frac{1}{4}}}$

Step 1.3.2

Factor $2.25$ out of $2.25$ .

$\frac{2.25 (1)}{4 x^{\frac{1}{4}}}$

Step 1.3.3

Factor $4$ out of $4 x^{\frac{1}{4}}$ .

$\frac{2.25 (1)}{4 (x^{\frac{1}{4}})}$

Step 1.3.4

Separate fractions.

$\frac{2.25}{4} \cdot \frac{1}{x^{\frac{1}{4}}}$

Step 1.3.5

Divide $2.25$ by $4$ .

$0.5625 \frac{1}{x^{\frac{1}{4}}}$

Step 1.3.6

Combine $0.5625$ and $\frac{1}{x^{\frac{1}{4}}}$ .

$\frac{0.5625}{x^{\frac{1}{4}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $0.5625$ is constant with respect to $x$ , the derivative of $\frac{0.5625}{x^{\frac{1}{4}}}$ with respect to $x$ is $0.5625 \frac{d}{d x} [\frac{1}{x^{\frac{1}{4}}}]$ .

$C'' (x) = 0.5625 \frac{d}{d x} (\frac{1}{x^{\frac{1}{4}}})$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

Rewrite $\frac{1}{x^{\frac{1}{4}}}$ as ${(x^{\frac{1}{4}})}^{- 1}$ .

$C'' (x) = 0.5625 \frac{d}{d x} ({(x^{\frac{1}{4}})}^{- 1})$

Step 2.2.2

Multiply the exponents in ${(x^{\frac{1}{4}})}^{- 1}$ .

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Step 2.2.2.1

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

$C'' (x) = 0.5625 \frac{d}{d x} (x^{\frac{1}{4} \cdot - 1})$

Step 2.2.2.2

Combine $\frac{1}{4}$ and $- 1$ .

$C'' (x) = 0.5625 \frac{d}{d x} (x^{\frac{- 1}{4}})$

Step 2.2.2.3

Move the negative in front of the fraction.

$C'' (x) = 0.5625 \frac{d}{d x} (x^{- \frac{1}{4}})$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{1}{4}$ .

$C'' (x) = 0.5625 (- \frac{1}{4} x^{- \frac{1}{4} - 1})$

Step 2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$C'' (x) = 0.5625 (- \frac{1}{4} x^{- \frac{1}{4} - 1 \cdot \frac{4}{4}})$

Step 2.5

Combine $- 1$ and $\frac{4}{4}$ .

$C'' (x) = 0.5625 (- \frac{1}{4} x^{- \frac{1}{4} + \frac{- 1 \cdot 4}{4}})$

Step 2.6

Combine the numerators over the common denominator.

$C'' (x) = 0.5625 (- \frac{1}{4} x^{\frac{- 1 - 1 \cdot 4}{4}})$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $4$ .

$C'' (x) = 0.5625 (- \frac{1}{4} x^{\frac{- 1 - 4}{4}})$

Step 2.7.2

Subtract $4$ from $- 1$ .

$C'' (x) = 0.5625 (- \frac{1}{4} x^{\frac{- 5}{4}})$

Step 2.8

Move the negative in front of the fraction.

$C'' (x) = 0.5625 (- \frac{1}{4} x^{- \frac{5}{4}})$

Step 2.9

Combine $x^{- \frac{5}{4}}$ and $\frac{1}{4}$ .

$C'' (x) = 0.5625 (- \frac{x^{- \frac{5}{4}}}{4})$

Step 2.10

Multiply $- 1$ by $0.5625$ .

$C'' (x) = - 0.5625 \frac{x^{- \frac{5}{4}}}{4}$

Step 2.11

Combine $- 0.5625$ and $\frac{x^{- \frac{5}{4}}}{4}$ .

$C'' (x) = \frac{- 0.5625 x^{- \frac{5}{4}}}{4}$

Step 2.12

Simplify the expression.

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Step 2.12.1

Move $x^{- \frac{5}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$C'' (x) = \frac{- 0.5625}{4 x^{\frac{5}{4}}}$

Step 2.12.2

Move the negative in front of the fraction.

$C'' (x) = - \frac{0.5625}{4 x^{\frac{5}{4}}}$

Step 3

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

$\frac{0.5625}{x^{\frac{1}{4}}} = 0$

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

Differentiate.

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Step 4.1.1.1

By the Sum Rule, the derivative of $375 + 0.75 x^{\frac{3}{4}}$ with respect to $x$ is $\frac{d}{d x} [375] + \frac{d}{d x} [0.75 x^{\frac{3}{4}}]$ .

$\frac{d}{d x} [375] + \frac{d}{d x} [0.75 x^{\frac{3}{4}}]$

Step 4.1.1.2

Since $375$ is constant with respect to $x$ , the derivative of $375$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [0.75 x^{\frac{3}{4}}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [0.75 x^{\frac{3}{4}}]$ .

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Step 4.1.2.1

Since $0.75$ is constant with respect to $x$ , the derivative of $0.75 x^{\frac{3}{4}}$ with respect to $x$ is $0.75 \frac{d}{d x} [x^{\frac{3}{4}}]$ .

$0 + 0.75 \frac{d}{d x} [x^{\frac{3}{4}}]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{4}$ .

$0 + 0.75 (\frac{3}{4} x^{\frac{3}{4} - 1})$

Step 4.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$0 + 0.75 (\frac{3}{4} x^{\frac{3}{4} - 1 \cdot \frac{4}{4}})$

Step 4.1.2.4

Combine $- 1$ and $\frac{4}{4}$ .

$0 + 0.75 (\frac{3}{4} x^{\frac{3}{4} + \frac{- 1 \cdot 4}{4}})$

Step 4.1.2.5

Combine the numerators over the common denominator.

$0 + 0.75 (\frac{3}{4} x^{\frac{3 - 1 \cdot 4}{4}})$

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 1$ by $4$ .

$0 + 0.75 (\frac{3}{4} x^{\frac{3 - 4}{4}})$

Step 4.1.2.6.2

Subtract $4$ from $3$ .

$0 + 0.75 (\frac{3}{4} x^{\frac{- 1}{4}})$

Step 4.1.2.7

Move the negative in front of the fraction.

$0 + 0.75 (\frac{3}{4} x^{- \frac{1}{4}})$

Step 4.1.2.8

Combine $\frac{3}{4}$ and $x^{- \frac{1}{4}}$ .

$0 + 0.75 \frac{3 x^{- \frac{1}{4}}}{4}$

Step 4.1.2.9

Combine $0.75$ and $\frac{3 x^{- \frac{1}{4}}}{4}$ .

$0 + \frac{0.75 (3 x^{- \frac{1}{4}})}{4}$

Step 4.1.2.10

Multiply $3$ by $0.75$ .

$0 + \frac{2.25 x^{- \frac{1}{4}}}{4}$

Step 4.1.2.11

Move $x^{- \frac{1}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 + \frac{2.25}{4 x^{\frac{1}{4}}}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Add $0$ and $\frac{2.25}{4 x^{\frac{1}{4}}}$ .

$\frac{2.25}{4 x^{\frac{1}{4}}}$

Step 4.1.3.2

Factor $2.25$ out of $2.25$ .

$\frac{2.25 (1)}{4 x^{\frac{1}{4}}}$

Step 4.1.3.3

Factor $4$ out of $4 x^{\frac{1}{4}}$ .

$\frac{2.25 (1)}{4 (x^{\frac{1}{4}})}$

Step 4.1.3.4

Separate fractions.

$\frac{2.25}{4} \cdot \frac{1}{x^{\frac{1}{4}}}$

Step 4.1.3.5

Divide $2.25$ by $4$ .

$0.5625 \frac{1}{x^{\frac{1}{4}}}$

Step 4.1.3.6

Combine $0.5625$ and $\frac{1}{x^{\frac{1}{4}}}$ .

$f' (x) = \frac{0.5625}{x^{\frac{1}{4}}}$

Step 4.2

The first derivative of $C (x)$ with respect to $x$ is $\frac{0.5625}{x^{\frac{1}{4}}}$ .

$\frac{0.5625}{x^{\frac{1}{4}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{0.5625}{x^{\frac{1}{4}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{0.5625}{x^{\frac{1}{4}}} = 0$

Step 5.2

Set the numerator equal to zero.

$0.5625 = 0$

Step 5.3

Since $0.5625 \neq 0$ , 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

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

$\frac{0.5625}{\sqrt[4]{x^{1}}}$

Step 6.1.2

Anything raised to $1$ is the base itself.

$\frac{0.5625}{\sqrt[4]{x}}$

Step 6.2

Set the denominator in $\frac{0.5625}{\sqrt[4]{x}}$ equal to $0$ to find where the expression is undefined.

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

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 $4$ .

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

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 $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${(x^{\frac{1}{4}})}^{4}$ .

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Step 6.3.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

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Step 6.3.2.2.1.1.1

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

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

Step 6.3.2.2.1.1.2

Cancel the common factor of $4$ .

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Step 6.3.2.2.1.1.2.1

Cancel the common factor.

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

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 6.3.2.2.1.2

Simplify.

$x = 0^{4}$

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 $0$ .

$x = 0$

Step 6.4

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

$x < 0$

Step 6.5

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.

$- \frac{0.5625}{4 {(0)}^{\frac{5}{4}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the expression.

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Step 9.1.1

Rewrite $0$ as $0^{4}$ .

$- \frac{0.5625}{4 {(0^{4})}^{\frac{5}{4}}}$

Step 9.1.2

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

$- \frac{0.5625}{4 \cdot 0^{4 (\frac{5}{4})}}$

Step 9.2

Cancel the common factor of $4$ .

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Step 9.2.1

Cancel the common factor.

$- \frac{0.5625}{4 \cdot 0^{4 (\frac{5}{4})}}$

Step 9.2.2

Rewrite the expression.

$- \frac{0.5625}{4 \cdot 0^{5}}$

Step 9.3

Simplify the expression.

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Step 9.3.1

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

$- \frac{0.5625}{4 \cdot 0}$

Step 9.3.2

Multiply $4$ by $0$ .

$- \frac{0.5625}{0}$

Step 9.3.3

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

Undefined

$- \frac{0.5625}{0}$

Step 9.4

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