Calculus Examples

$f (x) = (x - 1) (x^{2} - 8 x + 7)$

Step 1

Find the derivative.

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Step 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 $f (x) = x - 1$ and $g (x) = x^{2} - 8 x + 7$ .

$(x - 1) \frac{d}{d x} [x^{2} - 8 x + 7] + (x^{2} - 8 x + 7) \frac{d}{d x} [x - 1]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{2} - 8 x + 7$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [7]$ .

$(x - 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [7]) + (x^{2} - 8 x + 7) \frac{d}{d x} [x - 1]$

Step 1.2.2

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

$(x - 1) (2 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [7]) + (x^{2} - 8 x + 7) \frac{d}{d x} [x - 1]$

Step 1.2.3

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$(x - 1) (2 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [7]) + (x^{2} - 8 x + 7) \frac{d}{d x} [x - 1]$

Step 1.2.4

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

$(x - 1) (2 x - 8 \cdot 1 + \frac{d}{d x} [7]) + (x^{2} - 8 x + 7) \frac{d}{d x} [x - 1]$

Step 1.2.5

Multiply $- 8$ by $1$ .

$(x - 1) (2 x - 8 + \frac{d}{d x} [7]) + (x^{2} - 8 x + 7) \frac{d}{d x} [x - 1]$

Step 1.2.6

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$(x - 1) (2 x - 8 + 0) + (x^{2} - 8 x + 7) \frac{d}{d x} [x - 1]$

Step 1.2.7

Add $2 x - 8$ and $0$ .

$(x - 1) (2 x - 8) + (x^{2} - 8 x + 7) \frac{d}{d x} [x - 1]$

Step 1.2.8

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

$(x - 1) (2 x - 8) + (x^{2} - 8 x + 7) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 1.2.9

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

$(x - 1) (2 x - 8) + (x^{2} - 8 x + 7) (1 + \frac{d}{d x} [- 1])$

Step 1.2.10

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

$(x - 1) (2 x - 8) + (x^{2} - 8 x + 7) (1 + 0)$

Step 1.2.11

Simplify the expression.

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Step 1.2.11.1

Add $1$ and $0$ .

$(x - 1) (2 x - 8) + (x^{2} - 8 x + 7) \cdot 1$

Step 1.2.11.2

Multiply $x^{2} - 8 x + 7$ by $1$ .

$(x - 1) (2 x - 8) + x^{2} - 8 x + 7$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$(x - 1) (2 x) + (x - 1) \cdot - 8 + x^{2} - 8 x + 7$

Step 1.3.2

Apply the distributive property.

$x (2 x) - 1 (2 x) + (x - 1) \cdot - 8 + x^{2} - 8 x + 7$

Step 1.3.3

Apply the distributive property.

$x (2 x) - 1 (2 x) + x \cdot - 8 - 1 \cdot - 8 + x^{2} - 8 x + 7$

Step 1.3.4

Combine terms.

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Step 1.3.4.1

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

$2 (x^{1} x) - 1 (2 x) + x \cdot - 8 - 1 \cdot - 8 + x^{2} - 8 x + 7$

Step 1.3.4.2

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

$2 (x^{1} x^{1}) - 1 (2 x) + x \cdot - 8 - 1 \cdot - 8 + x^{2} - 8 x + 7$

Step 1.3.4.3

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

$2 x^{1 + 1} - 1 (2 x) + x \cdot - 8 - 1 \cdot - 8 + x^{2} - 8 x + 7$

Step 1.3.4.4

Add $1$ and $1$ .

$2 x^{2} - 1 (2 x) + x \cdot - 8 - 1 \cdot - 8 + x^{2} - 8 x + 7$

Step 1.3.4.5

Multiply $2$ by $- 1$ .

$2 x^{2} - 2 x + x \cdot - 8 - 1 \cdot - 8 + x^{2} - 8 x + 7$

Step 1.3.4.6

Move $- 8$ to the left of $x$ .

$2 x^{2} - 2 x - 8 \cdot x - 1 \cdot - 8 + x^{2} - 8 x + 7$

Step 1.3.4.7

Multiply $- 1$ by $- 8$ .

$2 x^{2} - 2 x - 8 x + 8 + x^{2} - 8 x + 7$

Step 1.3.4.8

Subtract $8 x$ from $- 2 x$ .

$2 x^{2} - 10 x + 8 + x^{2} - 8 x + 7$

Step 1.3.4.9

Add $2 x^{2}$ and $x^{2}$ .

$3 x^{2} - 10 x + 8 - 8 x + 7$

Step 1.3.4.10

Subtract $8 x$ from $- 10 x$ .

$3 x^{2} - 18 x + 8 + 7$

Step 1.3.4.11

Add $8$ and $7$ .

$3 x^{2} - 18 x + 15$

Step 2

Set the derivative equal to $0$ then solve the equation $3 x^{2} - 18 x + 15 = 0$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $3$ out of $3 x^{2} - 18 x + 15$ .

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Step 2.1.1.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) - 18 x + 15 = 0$

Step 2.1.1.2

Factor $3$ out of $- 18 x$ .

$3 (x^{2}) + 3 (- 6 x) + 15 = 0$

Step 2.1.1.3

Factor $3$ out of $15$ .

$3 x^{2} + 3 (- 6 x) + 3 \cdot 5 = 0$

Step 2.1.1.4

Factor $3$ out of $3 x^{2} + 3 (- 6 x)$ .

$3 (x^{2} - 6 x) + 3 \cdot 5 = 0$

Step 2.1.1.5

Factor $3$ out of $3 (x^{2} - 6 x) + 3 \cdot 5$ .

$3 (x^{2} - 6 x + 5) = 0$

Step 2.1.2

Factor.

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Step 2.1.2.1

Factor $x^{2} - 6 x + 5$ using the AC method.

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Step 2.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $5$ and whose sum is $- 6$ .

$- 5, - 1$

Step 2.1.2.1.2

Write the factored form using these integers.

$3 ((x - 5) (x - 1)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$3 (x - 5) (x - 1) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 5 = 0$

$x - 1 = 0$

Step 2.3

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.4

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.5

The final solution is all the values that make $3 (x - 5) (x - 1) = 0$ true.

$x = 5, 1$

Step 3

Solve the original function $f (x) = (x - 1) (x^{2} - 8 x + 7)$ at $x = 5$ .

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Step 3.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = ((5) - 1) ({(5)}^{2} - 8 \cdot 5 + 7)$

Step 3.2

Simplify the result.

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Step 3.2.1

Subtract $1$ from $5$ .

$f (5) = 4 ({(5)}^{2} - 8 \cdot 5 + 7)$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Raise $5$ to the power of $2$ .

$f (5) = 4 (25 - 8 \cdot 5 + 7)$

Step 3.2.2.2

Multiply $- 8$ by $5$ .

$f (5) = 4 (25 - 40 + 7)$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Subtract $40$ from $25$ .

$f (5) = 4 (- 15 + 7)$

Step 3.2.3.2

Add $- 15$ and $7$ .

$f (5) = 4 \cdot - 8$

Step 3.2.3.3

Multiply $4$ by $- 8$ .

$f (5) = - 32$

Step 3.2.4

The final answer is $- 32$ .

$- 32$

Step 4

Solve the original function $f (x) = (x - 1) (x^{2} - 8 x + 7)$ at $x = 1$ .

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Step 4.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = ((1) - 1) ({(1)}^{2} - 8 \cdot 1 + 7)$

Step 4.2

Simplify the result.

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Step 4.2.1

Subtract $1$ from $1$ .

$f (1) = 0 ({(1)}^{2} - 8 \cdot 1 + 7)$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

One to any power is one.

$f (1) = 0 (1 - 8 \cdot 1 + 7)$

Step 4.2.2.2

Multiply $- 8$ by $1$ .

$f (1) = 0 (1 - 8 + 7)$

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Subtract $8$ from $1$ .

$f (1) = 0 (- 7 + 7)$

Step 4.2.3.2

Add $- 7$ and $7$ .

$f (1) = 0 \cdot 0$

Step 4.2.3.3

Multiply $0$ by $0$ .

$f (1) = 0$

Step 4.2.4

The final answer is $0$ .

$0$

Step 5

The horizontal tangent lines on function $f (x) = (x - 1) (x^{2} - 8 x + 7)$ are $y = - 32, y = 0$ .

$y = - 32, y = 0$

Step 6