Calculus Examples

$x^{2} - 32 \sqrt{x}$

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

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{2} - 32 \sqrt{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 32 \sqrt{x}]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 32 \sqrt{x}]$

Step 1.1.1.2

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

$2 x + \frac{d}{d x} [- 32 \sqrt{x}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 32 \sqrt{x}]$ .

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Step 1.1.2.1

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

$2 x + \frac{d}{d x} [- 32 x^{\frac{1}{2}}]$

Step 1.1.2.2

Since $- 32$ is constant with respect to $x$ , the derivative of $- 32 x^{\frac{1}{2}}$ with respect to $x$ is $- 32 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$2 x - 32 \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 1.1.2.3

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

$2 x - 32 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 1.1.2.4

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

$2 x - 32 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 1.1.2.5

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

$2 x - 32 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 1.1.2.6

Combine the numerators over the common denominator.

$2 x - 32 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 1.1.2.7

Simplify the numerator.

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Step 1.1.2.7.1

Multiply $- 1$ by $2$ .

$2 x - 32 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 1.1.2.7.2

Subtract $2$ from $1$ .

$2 x - 32 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 1.1.2.8

Move the negative in front of the fraction.

$2 x - 32 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 1.1.2.9

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

$2 x - 32 \frac{x^{- \frac{1}{2}}}{2}$

Step 1.1.2.10

Combine $- 32$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$2 x + \frac{- 32 x^{- \frac{1}{2}}}{2}$

Step 1.1.2.11

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

$2 x + \frac{- 32}{2 x^{\frac{1}{2}}}$

Step 1.1.2.12

Factor $2$ out of $- 32$ .

$2 x + \frac{2 \cdot - 16}{2 x^{\frac{1}{2}}}$

Step 1.1.2.13

Cancel the common factors.

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Step 1.1.2.13.1

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

$2 x + \frac{2 \cdot - 16}{2 (x^{\frac{1}{2}})}$

Step 1.1.2.13.2

Cancel the common factor.

$2 x + \frac{2 \cdot - 16}{2 x^{\frac{1}{2}}}$

Step 1.1.2.13.3

Rewrite the expression.

$2 x + \frac{- 16}{x^{\frac{1}{2}}}$

Step 1.1.2.14

Move the negative in front of the fraction.

$f' (x) = 2 x - \frac{16}{x^{\frac{1}{2}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 x - \frac{16}{x^{\frac{1}{2}}}$ .

$2 x - \frac{16}{x^{\frac{1}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $2 x - \frac{16}{x^{\frac{1}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

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

Step 2.2.2

The LCM of one and any expression is the expression.

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

Step 2.3

Multiply each term in $2 x - \frac{16}{x^{\frac{1}{2}}} = 0$ by $x^{\frac{1}{2}}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $2 x - \frac{16}{x^{\frac{1}{2}}} = 0$ by $x^{\frac{1}{2}}$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

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

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Step 2.3.2.1.1.1

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

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

Step 2.3.2.1.1.2

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

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Step 2.3.2.1.1.2.1

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

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

Step 2.3.2.1.1.2.2

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

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

Step 2.3.2.1.1.3

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

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

Step 2.3.2.1.1.4

Combine the numerators over the common denominator.

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

Step 2.3.2.1.1.5

Add $1$ and $2$ .

$2 x^{\frac{3}{2}} - \frac{16}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 2.3.2.1.2

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

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Step 2.3.2.1.2.1

Move the leading negative in $- \frac{16}{x^{\frac{1}{2}}}$ into the numerator.

$2 x^{\frac{3}{2}} + \frac{- 16}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 2.3.2.1.2.2

Cancel the common factor.

$2 x^{\frac{3}{2}} + \frac{- 16}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 2.3.2.1.2.3

Rewrite the expression.

$2 x^{\frac{3}{2}} - 16 = 0 x^{\frac{1}{2}}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0$ by $x^{\frac{1}{2}}$ .

$2 x^{\frac{3}{2}} - 16 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Add $16$ to both sides of the equation.

$2 x^{\frac{3}{2}} = 16$

Step 2.4.2

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${(2 x^{\frac{3}{2}})}^{\frac{2}{3}} = 16^{\frac{2}{3}}$

Step 2.4.3

Simplify the left side.

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Step 2.4.3.1

Simplify ${(2 x^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 2.4.3.1.1

Apply the product rule to $2 x^{\frac{3}{2}}$ .

$2^{\frac{2}{3}} {(x^{\frac{3}{2}})}^{\frac{2}{3}} = 16^{\frac{2}{3}}$

Step 2.4.3.1.2

Multiply the exponents in ${(x^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 2.4.3.1.2.1

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

$2^{\frac{2}{3}} x^{\frac{3}{2} \cdot \frac{2}{3}} = 16^{\frac{2}{3}}$

Step 2.4.3.1.2.2

Cancel the common factor of $3$ .

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Step 2.4.3.1.2.2.1

Cancel the common factor.

$2^{\frac{2}{3}} x^{\frac{3}{2} \cdot \frac{2}{3}} = 16^{\frac{2}{3}}$

Step 2.4.3.1.2.2.2

Rewrite the expression.

$2^{\frac{2}{3}} x^{\frac{1}{2} \cdot 2} = 16^{\frac{2}{3}}$

Step 2.4.3.1.2.3

Cancel the common factor of $2$ .

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Step 2.4.3.1.2.3.1

Cancel the common factor.

$2^{\frac{2}{3}} x^{\frac{1}{2} \cdot 2} = 16^{\frac{2}{3}}$

Step 2.4.3.1.2.3.2

Rewrite the expression.

$2^{\frac{2}{3}} x^{1} = 16^{\frac{2}{3}}$

Step 2.4.3.1.3

Simplify.

$2^{\frac{2}{3}} x = 16^{\frac{2}{3}}$

Step 2.4.3.1.4

Reorder factors in $2^{\frac{2}{3}} x$ .

$x \cdot 2^{\frac{2}{3}} = 16^{\frac{2}{3}}$

Step 2.4.4

Divide each term in $x \cdot 2^{\frac{2}{3}} = 16^{\frac{2}{3}}$ by $2^{\frac{2}{3}}$ and simplify.

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Step 2.4.4.1

Divide each term in $x \cdot 2^{\frac{2}{3}} = 16^{\frac{2}{3}}$ by $2^{\frac{2}{3}}$ .

$\frac{x \cdot 2^{\frac{2}{3}}}{2^{\frac{2}{3}}} = \frac{16^{\frac{2}{3}}}{2^{\frac{2}{3}}}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor.

$\frac{x \cdot 2^{\frac{2}{3}}}{2^{\frac{2}{3}}} = \frac{16^{\frac{2}{3}}}{2^{\frac{2}{3}}}$

Step 2.4.4.2.2

Divide $x$ by $1$ .

$x = \frac{16^{\frac{2}{3}}}{2^{\frac{2}{3}}}$

Step 2.4.4.3

Simplify the right side.

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Step 2.4.4.3.1

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$x = {(\frac{16}{2})}^{\frac{2}{3}}$

Step 2.4.4.3.2

Simplify the expression.

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Step 2.4.4.3.2.1

Divide $16$ by $2$ .

$x = 8^{\frac{2}{3}}$

Step 2.4.4.3.2.2

Rewrite $8$ as $2^{3}$ .

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

Step 2.4.4.3.2.3

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

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

Step 2.4.4.3.3

Cancel the common factor of $3$ .

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Step 2.4.4.3.3.1

Cancel the common factor.

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

Step 2.4.4.3.3.2

Rewrite the expression.

$x = 2^{2}$

Step 2.4.4.3.4

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

$x = 4$

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.

$2 x - \frac{16}{\sqrt{x^{1}}}$

Step 3.1.2

Anything raised to $1$ is the base itself.

$2 x - \frac{16}{\sqrt{x}}$

Step 3.2

Set the denominator in $\frac{16}{\sqrt{x}}$ 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 radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 3.5

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 4

Evaluate $x^{2} - 32 \sqrt{x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 4$ .

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Step 4.1.1

Substitute $4$ for $x$ .

${(4)}^{2} - 32 \sqrt{4}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$16 - 32 \sqrt{4}$

Step 4.1.2.1.2

Rewrite $4$ as $2^{2}$ .

$16 - 32 \sqrt{2^{2}}$

Step 4.1.2.1.3

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

$16 - 32 \cdot 2$

Step 4.1.2.1.4

Multiply $- 32$ by $2$ .

$16 - 64$

Step 4.1.2.2

Subtract $64$ from $16$ .

$- 48$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

${(0)}^{2} - 32 \sqrt{0}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

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

$0 - 32 \sqrt{0}$

Step 4.2.2.1.2

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

$0 - 32 \sqrt{0^{2}}$

Step 4.2.2.1.3

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

$0 - 32 \cdot 0$

Step 4.2.2.1.4

Multiply $- 32$ by $0$ .

$0 + 0$

Step 4.2.2.2

Add $0$ and $0$ .

$0$

Step 4.3

List all of the points.

$(4, - 48), (0, 0)$

Step 5