Calculus Examples

$f (x) = - 4 x^{\frac{3}{2}} + 24 x + 13$

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

By the Sum Rule, the derivative of $- 4 x^{\frac{3}{2}} + 24 x + 13$ with respect to $x$ is $\frac{d}{d x} [- 4 x^{\frac{3}{2}}] + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$ .

$\frac{d}{d x} [- 4 x^{\frac{3}{2}}] + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 4 x^{\frac{3}{2}}]$ .

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Step 1.1.2.1

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

$- 4 \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.2

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

$- 4 (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.3

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

$- 4 (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.4

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

$- 4 (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.5

Combine the numerators over the common denominator.

$- 4 (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.6

Simplify the numerator.

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Step 1.1.2.6.1

Multiply $- 1$ by $2$ .

$- 4 (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.6.2

Subtract $2$ from $3$ .

$- 4 (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.7

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

$- 4 \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.8

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

$\frac{- 4 (3 x^{\frac{1}{2}})}{2} + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.9

Multiply $3$ by $- 4$ .

$\frac{- 12 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.10

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

$\frac{2 (- 6 x^{\frac{1}{2}})}{2} + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.11

Cancel the common factors.

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Step 1.1.2.11.1

Factor $2$ out of $2$ .

$\frac{2 (- 6 x^{\frac{1}{2}})}{2 (1)} + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.11.2

Cancel the common factor.

$\frac{2 (- 6 x^{\frac{1}{2}})}{2 \cdot 1} + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.11.3

Rewrite the expression.

$\frac{- 6 x^{\frac{1}{2}}}{1} + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.2.11.4

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

$- 6 x^{\frac{1}{2}} + \frac{d}{d x} [24 x] + \frac{d}{d x} [13]$

Step 1.1.3

Evaluate $\frac{d}{d x} [24 x]$ .

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Step 1.1.3.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x$ with respect to $x$ is $24 \frac{d}{d x} [x]$ .

$- 6 x^{\frac{1}{2}} + 24 \frac{d}{d x} [x] + \frac{d}{d x} [13]$

Step 1.1.3.2

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

$- 6 x^{\frac{1}{2}} + 24 \cdot 1 + \frac{d}{d x} [13]$

Step 1.1.3.3

Multiply $24$ by $1$ .

$- 6 x^{\frac{1}{2}} + 24 + \frac{d}{d x} [13]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $13$ is constant with respect to $x$ , the derivative of $13$ with respect to $x$ is $0$ .

$- 6 x^{\frac{1}{2}} + 24 + 0$

Step 1.1.4.2

Add $- 6 x^{\frac{1}{2}} + 24$ and $0$ .

$f' (x) = - 6 x^{\frac{1}{2}} + 24$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 6 x^{\frac{1}{2}} + 24$ .

$- 6 x^{\frac{1}{2}} + 24$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 6 x^{\frac{1}{2}} + 24 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 6 x^{\frac{1}{2}} + 24 = 0$

Step 2.2

Subtract $24$ from both sides of the equation.

$- 6 x^{\frac{1}{2}} = - 24$

Step 2.3

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

${(- 6 x^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

Step 2.4

Simplify the exponent.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Simplify ${(- 6 x^{\frac{1}{2}})}^{2}$ .

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Step 2.4.1.1.1

Apply the product rule to $- 6 x^{\frac{1}{2}}$ .

${(- 6)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

Step 2.4.1.1.2

Raise $- 6$ to the power of $2$ .

$36 {(x^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

Step 2.4.1.1.3

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

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Step 2.4.1.1.3.1

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

$36 x^{\frac{1}{2} \cdot 2} = {(- 24)}^{2}$

Step 2.4.1.1.3.2

Cancel the common factor of $2$ .

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Step 2.4.1.1.3.2.1

Cancel the common factor.

$36 x^{\frac{1}{2} \cdot 2} = {(- 24)}^{2}$

Step 2.4.1.1.3.2.2

Rewrite the expression.

$36 x^{1} = {(- 24)}^{2}$

Step 2.4.1.1.4

Simplify.

$36 x = {(- 24)}^{2}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Raise $- 24$ to the power of $2$ .

$36 x = 576$

Step 2.5

Divide each term in $36 x = 576$ by $36$ and simplify.

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Step 2.5.1

Divide each term in $36 x = 576$ by $36$ .

$\frac{36 x}{36} = \frac{576}{36}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $36$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{36 x}{36} = \frac{576}{36}$

Step 2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{576}{36}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Divide $576$ by $36$ .

$x = 16$

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.

$- 6 \sqrt{x^{1}} + 24$

Step 3.1.2

Anything raised to $1$ is the base itself.

$- 6 \sqrt{x} + 24$

Step 3.2

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

$x < 0$

Step 3.3

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 4

Evaluate $- 4 x^{\frac{3}{2}} + 24 x + 13$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 16$ .

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Step 4.1.1

Substitute $16$ for $x$ .

$- 4 {(16)}^{\frac{3}{2}} + 24 (16) + 13$

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

Rewrite $16$ as $4^{2}$ .

$- 4 {(4^{2})}^{\frac{3}{2}} + 24 (16) + 13$

Step 4.1.2.1.2

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

$- 4 \cdot 4^{2 (\frac{3}{2})} + 24 (16) + 13$

Step 4.1.2.1.3

Cancel the common factor of $2$ .

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Step 4.1.2.1.3.1

Cancel the common factor.

$- 4 \cdot 4^{2 (\frac{3}{2})} + 24 (16) + 13$

Step 4.1.2.1.3.2

Rewrite the expression.

$- 4 \cdot 4^{3} + 24 (16) + 13$

Step 4.1.2.1.4

Raise $4$ to the power of $3$ .

$- 4 \cdot 64 + 24 (16) + 13$

Step 4.1.2.1.5

Multiply $- 4$ by $64$ .

$- 256 + 24 (16) + 13$

Step 4.1.2.1.6

Multiply $24$ by $16$ .

$- 256 + 384 + 13$

Step 4.1.2.2

Simplify by adding numbers.

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Step 4.1.2.2.1

Add $- 256$ and $384$ .

$128 + 13$

Step 4.1.2.2.2

Add $128$ and $13$ .

$141$

Step 4.2

List all of the points.

$(16, 141)$

Step 5