Calculus Examples

$f (x) = 4 \sqrt{x^{3}} - 9$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $4 \sqrt{x^{3}} - 9$ with respect to $x$ is $\frac{d}{d x} [4 \sqrt{x^{3}}] + \frac{d}{d x} [- 9]$ .

$\frac{d}{d x} [4 \sqrt{x^{3}}] + \frac{d}{d x} [- 9]$

Step 1.2

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

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$\frac{d}{d x} [4 x^{\frac{3}{2}}] + \frac{d}{d x} [- 9]$

Step 1.2.2

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

$4 \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [- 9]$

Step 1.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$4 (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [- 9]$

Step 1.2.4

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

$4 (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 9]$

Step 1.2.5

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

$4 (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 9]$

Step 1.2.6

Combine the numerators over the common denominator.

$4 (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 9]$

Step 1.2.7

Simplify the numerator.

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Step 1.2.7.1

Multiply $- 1$ by $2$ .

$4 (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [- 9]$

Step 1.2.7.2

Subtract $2$ from $3$ .

$4 (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [- 9]$

Step 1.2.8

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

$4 \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [- 9]$

Step 1.2.9

Combine $4$ and $\frac{3 x^{\frac{1}{2}}}{2}$ .

$\frac{4 (3 x^{\frac{1}{2}})}{2} + \frac{d}{d x} [- 9]$

Step 1.2.10

Multiply $3$ by $4$ .

$\frac{12 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [- 9]$

Step 1.2.11

Factor $2$ out of $12 x^{\frac{1}{2}}$ .

$\frac{2 (6 x^{\frac{1}{2}})}{2} + \frac{d}{d x} [- 9]$

Step 1.2.12

Cancel the common factors.

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Step 1.2.12.1

Factor $2$ out of $2$ .

$\frac{2 (6 x^{\frac{1}{2}})}{2 (1)} + \frac{d}{d x} [- 9]$

Step 1.2.12.2

Cancel the common factor.

$\frac{2 (6 x^{\frac{1}{2}})}{2 \cdot 1} + \frac{d}{d x} [- 9]$

Step 1.2.12.3

Rewrite the expression.

$\frac{6 x^{\frac{1}{2}}}{1} + \frac{d}{d x} [- 9]$

Step 1.2.12.4

Divide $6 x^{\frac{1}{2}}$ by $1$ .

$6 x^{\frac{1}{2}} + \frac{d}{d x} [- 9]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$6 x^{\frac{1}{2}} + 0$

Step 1.3.2

Add $6 x^{\frac{1}{2}}$ and $0$ .

$6 x^{\frac{1}{2}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{\frac{1}{2}}$ with respect to $x$ is $6 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$f'' (x) = 6 \frac{d}{d x} (x^{\frac{1}{2}})$

Step 2.2

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

$f'' (x) = 6 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 2.3

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

$f'' (x) = 6 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 2.4

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

$f'' (x) = 6 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 2.5

Combine the numerators over the common denominator.

$f'' (x) = 6 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

$f'' (x) = 6 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 2.6.2

Subtract $2$ from $1$ .

$f'' (x) = 6 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 2.7

Move the negative in front of the fraction.

$f'' (x) = 6 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 2.8

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

$f'' (x) = 6 (\frac{x^{- \frac{1}{2}}}{2})$

Step 2.9

Combine $6$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$f'' (x) = \frac{6 x^{- \frac{1}{2}}}{2}$

Step 2.10

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

$f'' (x) = \frac{6}{2 x^{\frac{1}{2}}}$

Step 2.11

Factor $2$ out of $6$ .

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

Step 2.12

Cancel the common factors.

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Step 2.12.1

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

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

Step 2.12.2

Cancel the common factor.

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

Step 2.12.3

Rewrite the expression.

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

Step 3

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

$6 x^{\frac{1}{2}} = 0$

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

By the Sum Rule, the derivative of $4 \sqrt{x^{3}} - 9$ with respect to $x$ is $\frac{d}{d x} [4 \sqrt{x^{3}}] + \frac{d}{d x} [- 9]$ .

$\frac{d}{d x} [4 \sqrt{x^{3}}] + \frac{d}{d x} [- 9]$

Step 4.1.2

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

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Step 4.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$\frac{d}{d x} [4 x^{\frac{3}{2}}] + \frac{d}{d x} [- 9]$

Step 4.1.2.2

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

$4 \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [- 9]$

Step 4.1.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$4 (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [- 9]$

Step 4.1.2.4

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

$4 (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 9]$

Step 4.1.2.5

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

$4 (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 9]$

Step 4.1.2.6

Combine the numerators over the common denominator.

$4 (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 9]$

Step 4.1.2.7

Simplify the numerator.

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Step 4.1.2.7.1

Multiply $- 1$ by $2$ .

$4 (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [- 9]$

Step 4.1.2.7.2

Subtract $2$ from $3$ .

$4 (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [- 9]$

Step 4.1.2.8

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

$4 \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [- 9]$

Step 4.1.2.9

Combine $4$ and $\frac{3 x^{\frac{1}{2}}}{2}$ .

$\frac{4 (3 x^{\frac{1}{2}})}{2} + \frac{d}{d x} [- 9]$

Step 4.1.2.10

Multiply $3$ by $4$ .

$\frac{12 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [- 9]$

Step 4.1.2.11

Factor $2$ out of $12 x^{\frac{1}{2}}$ .

$\frac{2 (6 x^{\frac{1}{2}})}{2} + \frac{d}{d x} [- 9]$

Step 4.1.2.12

Cancel the common factors.

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Step 4.1.2.12.1

Factor $2$ out of $2$ .

$\frac{2 (6 x^{\frac{1}{2}})}{2 (1)} + \frac{d}{d x} [- 9]$

Step 4.1.2.12.2

Cancel the common factor.

$\frac{2 (6 x^{\frac{1}{2}})}{2 \cdot 1} + \frac{d}{d x} [- 9]$

Step 4.1.2.12.3

Rewrite the expression.

$\frac{6 x^{\frac{1}{2}}}{1} + \frac{d}{d x} [- 9]$

Step 4.1.2.12.4

Divide $6 x^{\frac{1}{2}}$ by $1$ .

$6 x^{\frac{1}{2}} + \frac{d}{d x} [- 9]$

Step 4.1.3

Differentiate using the Constant Rule.

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Step 4.1.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$6 x^{\frac{1}{2}} + 0$

Step 4.1.3.2

Add $6 x^{\frac{1}{2}}$ and $0$ .

$f' (x) = 6 x^{\frac{1}{2}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $6 x^{\frac{1}{2}}$ .

$6 x^{\frac{1}{2}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $6 x^{\frac{1}{2}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$6 x^{\frac{1}{2}} = 0$

Step 5.2

Divide each term in $6 x^{\frac{1}{2}} = 0$ by $6$ and simplify.

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Step 5.2.1

Divide each term in $6 x^{\frac{1}{2}} = 0$ by $6$ .

$\frac{6 x^{\frac{1}{2}}}{6} = \frac{0}{6}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor.

$\frac{6 x^{\frac{1}{2}}}{6} = \frac{0}{6}$

Step 5.2.2.2

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

$x^{\frac{1}{2}} = \frac{0}{6}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $0$ by $6$ .

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

Step 5.3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 5.4

Simplify the exponent.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

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

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Step 5.4.1.1.1

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

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Step 5.4.1.1.1.1

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

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

Step 5.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.4.1.1.1.2.1

Cancel the common factor.

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

Step 5.4.1.1.1.2.2

Rewrite the expression.

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

Step 5.4.1.1.2

Simplify.

$x = 0^{2}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

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

$x = 0$

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.

$6 \sqrt{x^{1}}$

Step 6.1.2

Anything raised to $1$ is the base itself.

$6 \sqrt{x}$

Step 6.2

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

$x < 0$

Step 6.3

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 $0$ .

$x < 0$

$(- \infty, 0)$

$x < 0$

$(- \infty, 0)$

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.

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

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 $0^{2}$ .

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

Step 9.1.2

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

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

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Cancel the common factor.

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

Step 9.2.2

Rewrite the expression.

$\frac{3}{0^{1}}$

Step 9.3

Evaluate the exponent.

$\frac{3}{0}$

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