Calculus Examples

$f (x) = 15 - \frac{3}{2} x^{\frac{1}{2}}$

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 $15 - \frac{3}{2} x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [15] + \frac{d}{d x} [- \frac{3}{2} x^{\frac{1}{2}}]$ .

$\frac{d}{d x} [15] + \frac{d}{d x} [- \frac{3}{2} x^{\frac{1}{2}}]$

Step 1.1.2

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{3}{2} x^{\frac{1}{2}}]$

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{3}{2} x^{\frac{1}{2}}]$ .

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Step 1.2.1

Since $- \frac{3}{2}$ is constant with respect to $x$ , the derivative of $- \frac{3}{2} x^{\frac{1}{2}}$ with respect to $x$ is $- \frac{3}{2} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$0 - \frac{3}{2} \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Combine the numerators over the common denominator.

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

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $2$ .

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

Step 1.2.6.2

Subtract $2$ from $1$ .

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

Step 1.2.7

Move the negative in front of the fraction.

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

Step 1.2.8

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

$0 - \frac{3}{2} \cdot \frac{x^{- \frac{1}{2}}}{2}$

Step 1.2.9

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

$0 - \frac{x^{- \frac{1}{2}} \cdot 3}{2 \cdot 2}$

Step 1.2.10

Multiply $2$ by $2$ .

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

Step 1.2.11

Move $3$ to the left of $x^{- \frac{1}{2}}$ .

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

Step 1.2.12

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

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

Step 1.3

Subtract $\frac{3}{4 x^{\frac{1}{2}}}$ from $0$ .

$- \frac{3}{4 x^{\frac{1}{2}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- \frac{3}{4}$ is constant with respect to $x$ , the derivative of $- \frac{3}{4 x^{\frac{1}{2}}}$ with respect to $x$ is $- \frac{3}{4} \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

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

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

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

Step 2.7.2

Subtract $2$ from $- 1$ .

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

Step 2.8

Move the negative in front of the fraction.

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

Step 2.9

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

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

Step 2.10

Multiply.

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Step 2.10.1

Multiply $- 1$ by $- 1$ .

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

Step 2.10.2

Multiply $\frac{3}{4}$ by $1$ .

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

Step 2.11

Multiply $\frac{3}{4}$ by $\frac{x^{- \frac{3}{2}}}{2}$ .

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

Step 2.12

Multiply.

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Step 2.12.1

Multiply $4$ by $2$ .

$f'' (x) = \frac{3 x^{- \frac{3}{2}}}{8}$

Step 2.12.2

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

$f'' (x) = \frac{3}{8 x^{\frac{3}{2}}}$

Step 3

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

$- \frac{3}{4 x^{\frac{1}{2}}} = 0$

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 $15 - \frac{3}{2} x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [15] + \frac{d}{d x} [- \frac{3}{2} x^{\frac{1}{2}}]$ .

$\frac{d}{d x} [15] + \frac{d}{d x} [- \frac{3}{2} x^{\frac{1}{2}}]$

Step 4.1.1.2

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{3}{2} x^{\frac{1}{2}}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [- \frac{3}{2} x^{\frac{1}{2}}]$ .

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Step 4.1.2.1

Since $- \frac{3}{2}$ is constant with respect to $x$ , the derivative of $- \frac{3}{2} x^{\frac{1}{2}}$ with respect to $x$ is $- \frac{3}{2} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$0 - \frac{3}{2} \frac{d}{d x} [x^{\frac{1}{2}}]$

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 $n = \frac{1}{2}$ .

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

Step 4.1.2.3

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

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

Step 4.1.2.4

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

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

Step 4.1.2.5

Combine the numerators over the common denominator.

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

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 1$ by $2$ .

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

Step 4.1.2.6.2

Subtract $2$ from $1$ .

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

Step 4.1.2.7

Move the negative in front of the fraction.

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

Step 4.1.2.8

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

$0 - \frac{3}{2} \cdot \frac{x^{- \frac{1}{2}}}{2}$

Step 4.1.2.9

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

$0 - \frac{x^{- \frac{1}{2}} \cdot 3}{2 \cdot 2}$

Step 4.1.2.10

Multiply $2$ by $2$ .

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

Step 4.1.2.11

Move $3$ to the left of $x^{- \frac{1}{2}}$ .

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

Step 4.1.2.12

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

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

Step 4.1.3

Subtract $\frac{3}{4 x^{\frac{1}{2}}}$ from $0$ .

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{3}{4 x^{\frac{1}{2}}}$ .

$- \frac{3}{4 x^{\frac{1}{2}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{3}{4 x^{\frac{1}{2}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{3}{4 x^{\frac{1}{2}}} = 0$

Step 5.2

Set the numerator equal to zero.

$3 = 0$

Step 5.3

Since $3 \neq 0$ , 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.

$- \frac{3}{4 \sqrt{x^{1}}}$

Step 6.1.2

Anything raised to $1$ is the base itself.

$- \frac{3}{4 \sqrt{x}}$

Step 6.2

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

$4 \sqrt{x} = 0$

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, square both sides of the equation.

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

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${(4 x^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1

Apply the product rule to $4 x^{\frac{1}{2}}$ .

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

Step 6.3.2.2.1.2

Raise $4$ to the power of $2$ .

$16 {(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2.1.3

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

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Step 6.3.2.2.1.3.1

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

$16 x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.3.2.1

Cancel the common factor.

$16 x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.3.2.2

Rewrite the expression.

$16 x^{1} = 0^{2}$

Step 6.3.2.2.1.4

Simplify.

$16 x = 0^{2}$

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 $0$ .

$16 x = 0$

Step 6.3.3

Divide each term in $16 x = 0$ by $16$ and simplify.

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Step 6.3.3.1

Divide each term in $16 x = 0$ by $16$ .

$\frac{16 x}{16} = \frac{0}{16}$

Step 6.3.3.2

Simplify the left side.

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Step 6.3.3.2.1

Cancel the common factor of $16$ .

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Step 6.3.3.2.1.1

Cancel the common factor.

$\frac{16 x}{16} = \frac{0}{16}$

Step 6.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{16}$

Step 6.3.3.3

Simplify the right side.

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Step 6.3.3.3.1

Divide $0$ by $16$ .

$x = 0$

Step 6.4

Set the radicand in $\sqrt{x}$ 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.

$\frac{3}{8 {(0)}^{\frac{3}{2}}}$

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 $0^{2}$ .

$\frac{3}{8 {(0^{2})}^{\frac{3}{2}}}$

Step 9.1.2

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

$\frac{3}{8 \cdot 0^{2 (\frac{3}{2})}}$

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Cancel the common factor.

$\frac{3}{8 \cdot 0^{2 (\frac{3}{2})}}$

Step 9.2.2

Rewrite the expression.

$\frac{3}{8 \cdot 0^{3}}$

Step 9.3

Simplify the expression.

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Step 9.3.1

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

$\frac{3}{8 \cdot 0}$

Step 9.3.2

Multiply $8$ by $0$ .

$\frac{3}{0}$

Step 9.3.3

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

Undefined

$\frac{3}{0}$

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