Calculus Examples

$f (x) = (4 x - 6) (5 x + 1)$

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 using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 4 x - 6$ and $g (x) = 5 x + 1$ .

$(4 x - 6) \frac{d}{d x} [5 x + 1] + (5 x + 1) \frac{d}{d x} [4 x - 6]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $5 x + 1$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [1]$ .

$(4 x - 6) (\frac{d}{d x} [5 x] + \frac{d}{d x} [1]) + (5 x + 1) \frac{d}{d x} [4 x - 6]$

Step 1.1.2.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$(4 x - 6) (5 \frac{d}{d x} [x] + \frac{d}{d x} [1]) + (5 x + 1) \frac{d}{d x} [4 x - 6]$

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 $n = 1$ .

$(4 x - 6) (5 \cdot 1 + \frac{d}{d x} [1]) + (5 x + 1) \frac{d}{d x} [4 x - 6]$

Step 1.1.2.4

Multiply $5$ by $1$ .

$(4 x - 6) (5 + \frac{d}{d x} [1]) + (5 x + 1) \frac{d}{d x} [4 x - 6]$

Step 1.1.2.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(4 x - 6) (5 + 0) + (5 x + 1) \frac{d}{d x} [4 x - 6]$

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $5$ and $0$ .

$(4 x - 6) \cdot 5 + (5 x + 1) \frac{d}{d x} [4 x - 6]$

Step 1.1.2.6.2

Move $5$ to the left of $4 x - 6$ .

$5 \cdot (4 x - 6) + (5 x + 1) \frac{d}{d x} [4 x - 6]$

Step 1.1.2.7

By the Sum Rule, the derivative of $4 x - 6$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 6]$ .

$5 (4 x - 6) + (5 x + 1) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 6])$

Step 1.1.2.8

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

$5 (4 x - 6) + (5 x + 1) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 6])$

Step 1.1.2.9

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

$5 (4 x - 6) + (5 x + 1) (4 \cdot 1 + \frac{d}{d x} [- 6])$

Step 1.1.2.10

Multiply $4$ by $1$ .

$5 (4 x - 6) + (5 x + 1) (4 + \frac{d}{d x} [- 6])$

Step 1.1.2.11

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$5 (4 x - 6) + (5 x + 1) (4 + 0)$

Step 1.1.2.12

Simplify the expression.

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Step 1.1.2.12.1

Add $4$ and $0$ .

$5 (4 x - 6) + (5 x + 1) \cdot 4$

Step 1.1.2.12.2

Move $4$ to the left of $5 x + 1$ .

$5 (4 x - 6) + 4 \cdot (5 x + 1)$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$5 (4 x) + 5 \cdot - 6 + 4 (5 x + 1)$

Step 1.1.3.2

Apply the distributive property.

$5 (4 x) + 5 \cdot - 6 + 4 (5 x) + 4 \cdot 1$

Step 1.1.3.3

Combine terms.

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Step 1.1.3.3.1

Multiply $4$ by $5$ .

$20 x + 5 \cdot - 6 + 4 (5 x) + 4 \cdot 1$

Step 1.1.3.3.2

Multiply $5$ by $- 6$ .

$20 x - 30 + 4 (5 x) + 4 \cdot 1$

Step 1.1.3.3.3

Multiply $5$ by $4$ .

$20 x - 30 + 20 x + 4 \cdot 1$

Step 1.1.3.3.4

Multiply $4$ by $1$ .

$20 x - 30 + 20 x + 4$

Step 1.1.3.3.5

Add $20 x$ and $20 x$ .

$40 x - 30 + 4$

Step 1.1.3.3.6

Add $- 30$ and $4$ .

$f' (x) = 40 x - 26$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $40 x - 26$ .

$40 x - 26$

Step 2

Set the first derivative equal to $0$ then solve the equation $40 x - 26 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$40 x - 26 = 0$

Step 2.2

Add $26$ to both sides of the equation.

$40 x = 26$

Step 2.3

Divide each term in $40 x = 26$ by $40$ and simplify.

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Step 2.3.1

Divide each term in $40 x = 26$ by $40$ .

$\frac{40 x}{40} = \frac{26}{40}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $40$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{40 x}{40} = \frac{26}{40}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{26}{40}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $26$ and $40$ .

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Step 2.3.3.1.1

Factor $2$ out of $26$ .

$x = \frac{2 (13)}{40}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $2$ out of $40$ .

$x = \frac{2 \cdot 13}{2 \cdot 20}$

Step 2.3.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 13}{2 \cdot 20}$

Step 2.3.3.1.2.3

Rewrite the expression.

$x = \frac{13}{20}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $(4 x - 6) (5 x + 1)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{13}{20}$ .

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Step 4.1.1

Substitute $\frac{13}{20}$ for $x$ .

$(4 (\frac{13}{20}) - 6) (5 (\frac{13}{20}) + 1)$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Cancel the common factor of $4$ .

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Step 4.1.2.1.1

Factor $4$ out of $20$ .

$(4 \frac{13}{4 (5)} - 6) (5 (\frac{13}{20}) + 1)$

Step 4.1.2.1.2

Cancel the common factor.

$(4 \frac{13}{4 \cdot 5} - 6) (5 (\frac{13}{20}) + 1)$

Step 4.1.2.1.3

Rewrite the expression.

$(\frac{13}{5} - 6) (5 (\frac{13}{20}) + 1)$

Step 4.1.2.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$(\frac{13}{5} - 6 \cdot \frac{5}{5}) (5 (\frac{13}{20}) + 1)$

Step 4.1.2.3

Combine $- 6$ and $\frac{5}{5}$ .

$(\frac{13}{5} + \frac{- 6 \cdot 5}{5}) (5 (\frac{13}{20}) + 1)$

Step 4.1.2.4

Combine the numerators over the common denominator.

$\frac{13 - 6 \cdot 5}{5} (5 (\frac{13}{20}) + 1)$

Step 4.1.2.5

Simplify the numerator.

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Step 4.1.2.5.1

Multiply $- 6$ by $5$ .

$\frac{13 - 30}{5} (5 (\frac{13}{20}) + 1)$

Step 4.1.2.5.2

Subtract $30$ from $13$ .

$\frac{- 17}{5} (5 (\frac{13}{20}) + 1)$

Step 4.1.2.6

Move the negative in front of the fraction.

$- \frac{17}{5} (5 (\frac{13}{20}) + 1)$

Step 4.1.2.7

Cancel the common factor of $5$ .

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Step 4.1.2.7.1

Factor $5$ out of $20$ .

$- \frac{17}{5} (5 \frac{13}{5 (4)} + 1)$

Step 4.1.2.7.2

Cancel the common factor.

$- \frac{17}{5} (5 \frac{13}{5 \cdot 4} + 1)$

Step 4.1.2.7.3

Rewrite the expression.

$- \frac{17}{5} (\frac{13}{4} + 1)$

Step 4.1.2.8

Simplify the expression.

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Step 4.1.2.8.1

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

$- \frac{17}{5} (\frac{13}{4} + \frac{4}{4})$

Step 4.1.2.8.2

Combine the numerators over the common denominator.

$- \frac{17}{5} \cdot \frac{13 + 4}{4}$

Step 4.1.2.8.3

Add $13$ and $4$ .

$- \frac{17}{5} \cdot \frac{17}{4}$

Step 4.1.2.9

Multiply $- \frac{17}{5} \cdot \frac{17}{4}$ .

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Step 4.1.2.9.1

Multiply $\frac{17}{4}$ by $\frac{17}{5}$ .

$- \frac{17 \cdot 17}{4 \cdot 5}$

Step 4.1.2.9.2

Multiply $17$ by $17$ .

$- \frac{289}{4 \cdot 5}$

Step 4.1.2.9.3

Multiply $4$ by $5$ .

$- \frac{289}{20}$

Step 4.2

List all of the points.

$(\frac{13}{20}, - \frac{289}{20})$

Step 5