Calculus Examples

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

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

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

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

Step 1.1.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.1.3.1.2

Multiply $x$ by $1$ .

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

Step 1.1.3.1.3

Multiply $x$ by $1$ .

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

Step 1.1.3.1.4

Multiply $1$ by $1$ .

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

Step 1.1.3.2

Add $x$ and $x$ .

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

Step 1.1.4

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

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

Step 1.1.5

Differentiate.

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Step 1.1.5.1

By the Sum Rule, the derivative of $x^{2} + 2 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

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

Step 1.1.5.2

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

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

Step 1.1.5.3

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

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

Step 1.1.5.4

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

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

Step 1.1.5.5

Multiply $2$ by $1$ .

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

Step 1.1.5.6

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

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

Step 1.1.5.7

Add $2 x + 2$ and $0$ .

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

Step 1.1.5.8

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

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

Step 1.1.5.9

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

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

Step 1.1.5.10

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 1.1.5.11

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

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

Step 1.1.5.12

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

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

Step 1.1.5.13

Simplify the expression.

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Step 1.1.5.13.1

Multiply $- 1$ by $1$ .

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

Step 1.1.5.13.2

Move $- 1$ to the left of $x^{2} + 2 x + 1$ .

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

Step 1.1.5.13.3

Rewrite $- 1 (x^{2} + 2 x + 1)$ as $- (x^{2} + 2 x + 1)$ .

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

Step 1.1.6

Simplify.

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Step 1.1.6.1

Apply the distributive property.

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

Step 1.1.6.2

Apply the distributive property.

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

Step 1.1.6.3

Apply the distributive property.

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

Step 1.1.6.4

Apply the distributive property.

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

Step 1.1.6.5

Combine terms.

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Step 1.1.6.5.1

Multiply $2$ by $8$ .

$16 x - x (2 x) + 8 \cdot 2 - x \cdot 2 - x^{2} - (2 x) - 1 \cdot 1$

Step 1.1.6.5.2

Multiply $2$ by $- 1$ .

$16 x - 2 x \cdot x + 8 \cdot 2 - x \cdot 2 - x^{2} - (2 x) - 1 \cdot 1$

Step 1.1.6.5.3

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

$16 x - 2 (x^{1} x) + 8 \cdot 2 - x \cdot 2 - x^{2} - (2 x) - 1 \cdot 1$

Step 1.1.6.5.4

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

$16 x - 2 (x^{1} x^{1}) + 8 \cdot 2 - x \cdot 2 - x^{2} - (2 x) - 1 \cdot 1$

Step 1.1.6.5.5

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

$16 x - 2 x^{1 + 1} + 8 \cdot 2 - x \cdot 2 - x^{2} - (2 x) - 1 \cdot 1$

Step 1.1.6.5.6

Add $1$ and $1$ .

$16 x - 2 x^{2} + 8 \cdot 2 - x \cdot 2 - x^{2} - (2 x) - 1 \cdot 1$

Step 1.1.6.5.7

Multiply $8$ by $2$ .

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

Step 1.1.6.5.8

Multiply $2$ by $- 1$ .

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

Step 1.1.6.5.9

Subtract $2 x$ from $16 x$ .

$14 x - 2 x^{2} + 16 - x^{2} - (2 x) - 1 \cdot 1$

Step 1.1.6.5.10

Multiply $2$ by $- 1$ .

$14 x - 2 x^{2} + 16 - x^{2} - 2 x - 1 \cdot 1$

Step 1.1.6.5.11

Multiply $- 1$ by $1$ .

$14 x - 2 x^{2} + 16 - x^{2} - 2 x - 1$

Step 1.1.6.5.12

Subtract $x^{2}$ from $- 2 x^{2}$ .

$14 x - 3 x^{2} + 16 - 2 x - 1$

Step 1.1.6.5.13

Subtract $2 x$ from $14 x$ .

$12 x - 3 x^{2} + 16 - 1$

Step 1.1.6.5.14

Subtract $1$ from $16$ .

$12 x - 3 x^{2} + 15$

Step 1.1.6.6

Reorder terms.

$f' (x) = - 3 x^{2} + 12 x + 15$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 3 x^{2} + 12 x + 15$ .

$- 3 x^{2} + 12 x + 15$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 3 x^{2} + 12 x + 15 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 3 x^{2} + 12 x + 15 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $- 3$ out of $- 3 x^{2} + 12 x + 15$ .

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Step 2.2.1.1

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

$- 3 x^{2} + 12 x + 15 = 0$

Step 2.2.1.2

Factor $- 3$ out of $12 x$ .

$- 3 x^{2} - 3 (- 4 x) + 15 = 0$

Step 2.2.1.3

Factor $- 3$ out of $15$ .

$- 3 x^{2} - 3 (- 4 x) - 3 \cdot - 5 = 0$

Step 2.2.1.4

Factor $- 3$ out of $- 3 (x^{2}) - 3 (- 4 x)$ .

$- 3 (x^{2} - 4 x) - 3 \cdot - 5 = 0$

Step 2.2.1.5

Factor $- 3$ out of $- 3 (x^{2} - 4 x) - 3 (- 5)$ .

$- 3 (x^{2} - 4 x - 5) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $x^{2} - 4 x - 5$ using the AC method.

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Step 2.2.2.1.1

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 $- 4$ .

$- 5, 1$

Step 2.2.2.1.2

Write the factored form using these integers.

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

Step 2.2.2.2

Remove unnecessary parentheses.

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

Step 2.3

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 2.4

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

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Step 2.4.1

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

$x - 5 = 0$

Step 2.4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.5

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

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Step 2.5.1

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

$x + 1 = 0$

Step 2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.6

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

$x = 5, - 1$

Step 3

The values which make the derivative equal to $0$ are $5, - 1$ .

$5, - 1$

Step 4

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 1) \cup (- 1, 5) \cup (5, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = - 3 {(- 2)}^{2} + 12 (- 2) + 15$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f' (- 2) = - 3 \cdot 4 + 12 (- 2) + 15$

Step 5.2.1.2

Multiply $- 3$ by $4$ .

$f' (- 2) = - 12 + 12 (- 2) + 15$

Step 5.2.1.3

Multiply $12$ by $- 2$ .

$f' (- 2) = - 12 - 24 + 15$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $24$ from $- 12$ .

$f' (- 2) = - 36 + 15$

Step 5.2.2.2

Add $- 36$ and $15$ .

$f' (- 2) = - 21$

Step 5.2.3

The final answer is $- 21$ .

$- 21$

Step 5.3

At $x = - 2$ the derivative is $- 21$ . Since this is negative, the function is decreasing on $(- \infty, - 1)$ .

Decreasing on $(- \infty, - 1)$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(- 1, 5)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = - 3 {(2)}^{2} + 12 (2) + 15$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

$f' (2) = - 3 \cdot 4 + 12 (2) + 15$

Step 6.2.1.2

Multiply $- 3$ by $4$ .

$f' (2) = - 12 + 12 (2) + 15$

Step 6.2.1.3

Multiply $12$ by $2$ .

$f' (2) = - 12 + 24 + 15$

Step 6.2.2

Simplify by adding numbers.

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Step 6.2.2.1

Add $- 12$ and $24$ .

$f' (2) = 12 + 15$

Step 6.2.2.2

Add $12$ and $15$ .

$f' (2) = 27$

Step 6.2.3

The final answer is $27$ .

$27$

Step 6.3

At $x = 2$ the derivative is $27$ . Since this is positive, the function is increasing on $(- 1, 5)$ .

Increasing on $(- 1, 5)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(5, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $6$ in the expression.

$f' (6) = - 3 {(6)}^{2} + 12 (6) + 15$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

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

$f' (6) = - 3 \cdot 36 + 12 (6) + 15$

Step 7.2.1.2

Multiply $- 3$ by $36$ .

$f' (6) = - 108 + 12 (6) + 15$

Step 7.2.1.3

Multiply $12$ by $6$ .

$f' (6) = - 108 + 72 + 15$

Step 7.2.2

Simplify by adding numbers.

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Step 7.2.2.1

Add $- 108$ and $72$ .

$f' (6) = - 36 + 15$

Step 7.2.2.2

Add $- 36$ and $15$ .

$f' (6) = - 21$

Step 7.2.3

The final answer is $- 21$ .

$- 21$

Step 7.3

At $x = 6$ the derivative is $- 21$ . Since this is negative, the function is decreasing on $(5, \infty)$ .

Decreasing on $(5, \infty)$ since $f' (x) < 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- 1, 5)$

Decreasing on: $(- \infty, - 1), (5, \infty)$

Step 9