Calculus Examples

${(x + 1)}^{5} - 5 x - 2$

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

By the Sum Rule, the derivative of ${(x + 1)}^{5} - 5 x - 2$ with respect to $x$ is $\frac{d}{d x} [{(x + 1)}^{5}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 2]$ .

$f' (x) = \frac{d}{d x} ({(x + 1)}^{5}) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (- 2)$

Step 1.1.2

Evaluate $\frac{d}{d x} [{(x + 1)}^{5}]$ .

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Step 1.1.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x + 1$ .

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Step 1.1.2.1.1

To apply the Chain Rule, set $u$ as $x + 1$ .

$f' (x) = \frac{d}{d u} (u^{5}) \frac{d}{d x} (x + 1) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (- 2)$

Step 1.1.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$f' (x) = 5 u^{4} \frac{d}{d x} (x + 1) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (- 2)$

Step 1.1.2.1.3

Replace all occurrences of $u$ with $x + 1$ .

$f' (x) = 5 {(x + 1)}^{4} \frac{d}{d x} (x + 1) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (- 2)$

Step 1.1.2.2

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

$f' (x) = 5 {(x + 1)}^{4} (\frac{d}{d x} (x) + \frac{d}{d x} (1)) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (- 2)$

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 $n = 1$ .

$f' (x) = 5 {(x + 1)}^{4} (1 + \frac{d}{d x} (1)) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (- 2)$

Step 1.1.2.4

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

$f' (x) = 5 {(x + 1)}^{4} (1 + 0) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (- 2)$

Step 1.1.2.5

Add $1$ and $0$ .

$f' (x) = 5 {(x + 1)}^{4} \cdot 1 + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (- 2)$

Step 1.1.2.6

Multiply $5$ by $1$ .

$f' (x) = 5 {(x + 1)}^{4} + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (- 2)$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 1.1.3.1

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

$f' (x) = 5 {(x + 1)}^{4} - 5 \frac{d}{d x} x + \frac{d}{d x} (- 2)$

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 $n = 1$ .

$f' (x) = 5 {(x + 1)}^{4} - 5 \cdot 1 + \frac{d}{d x} (- 2)$

Step 1.1.3.3

Multiply $- 5$ by $1$ .

$f' (x) = 5 {(x + 1)}^{4} - 5 + \frac{d}{d x} (- 2)$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$f' (x) = 5 {(x + 1)}^{4} - 5 + 0$

Step 1.1.4.2

Add $5 {(x + 1)}^{4} - 5$ and $0$ .

$f' (x) = 5 {(x + 1)}^{4} - 5$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $5 {(x + 1)}^{4} - 5$ .

$5 {(x + 1)}^{4} - 5$

Step 2

Set the first derivative equal to $0$ then solve the equation $5 {(x + 1)}^{4} - 5 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$5 {(x + 1)}^{4} - 5 = 0$

Step 2.2

Simplify $5 {(x + 1)}^{4} - 5$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Use the Binomial Theorem.

$5 (x^{4} + 4 x^{3} \cdot 1 + 6 x^{2} \cdot 1^{2} + 4 x \cdot 1^{3} + 1^{4}) - 5 = 0$

Step 2.2.1.2

Simplify each term.

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Step 2.2.1.2.1

Multiply $4$ by $1$ .

$5 (x^{4} + 4 x^{3} + 6 x^{2} \cdot 1^{2} + 4 x \cdot 1^{3} + 1^{4}) - 5 = 0$

Step 2.2.1.2.2

One to any power is one.

$5 (x^{4} + 4 x^{3} + 6 x^{2} \cdot 1 + 4 x \cdot 1^{3} + 1^{4}) - 5 = 0$

Step 2.2.1.2.3

Multiply $6$ by $1$ .

$5 (x^{4} + 4 x^{3} + 6 x^{2} + 4 x \cdot 1^{3} + 1^{4}) - 5 = 0$

Step 2.2.1.2.4

One to any power is one.

$5 (x^{4} + 4 x^{3} + 6 x^{2} + 4 x \cdot 1 + 1^{4}) - 5 = 0$

Step 2.2.1.2.5

Multiply $4$ by $1$ .

$5 (x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1^{4}) - 5 = 0$

Step 2.2.1.2.6

One to any power is one.

$5 (x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1) - 5 = 0$

Step 2.2.1.3

Apply the distributive property.

$5 x^{4} + 5 (4 x^{3}) + 5 (6 x^{2}) + 5 (4 x) + 5 \cdot 1 - 5 = 0$

Step 2.2.1.4

Simplify.

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Step 2.2.1.4.1

Multiply $4$ by $5$ .

$5 x^{4} + 20 x^{3} + 5 (6 x^{2}) + 5 (4 x) + 5 \cdot 1 - 5 = 0$

Step 2.2.1.4.2

Multiply $6$ by $5$ .

$5 x^{4} + 20 x^{3} + 30 x^{2} + 5 (4 x) + 5 \cdot 1 - 5 = 0$

Step 2.2.1.4.3

Multiply $4$ by $5$ .

$5 x^{4} + 20 x^{3} + 30 x^{2} + 20 x + 5 \cdot 1 - 5 = 0$

Step 2.2.1.4.4

Multiply $5$ by $1$ .

$5 x^{4} + 20 x^{3} + 30 x^{2} + 20 x + 5 - 5 = 0$

Step 2.2.2

Combine the opposite terms in $5 x^{4} + 20 x^{3} + 30 x^{2} + 20 x + 5 - 5$ .

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Step 2.2.2.1

Subtract $5$ from $5$ .

$5 x^{4} + 20 x^{3} + 30 x^{2} + 20 x + 0 = 0$

Step 2.2.2.2

Add $5 x^{4} + 20 x^{3} + 30 x^{2} + 20 x$ and $0$ .

$5 x^{4} + 20 x^{3} + 30 x^{2} + 20 x = 0$

Step 2.3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = - 2, 0$

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 ${(x + 1)}^{5} - 5 x - 2$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 2$ .

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Step 4.1.1

Substitute $- 2$ for $x$ .

${((- 2) + 1)}^{5} - 5 \cdot - 2 - 2$

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

Add $- 2$ and $1$ .

${(- 1)}^{5} - 5 \cdot - 2 - 2$

Step 4.1.2.1.2

Raise $- 1$ to the power of $5$ .

$- 1 - 5 \cdot - 2 - 2$

Step 4.1.2.1.3

Multiply $- 5$ by $- 2$ .

$- 1 + 10 - 2$

Step 4.1.2.2

Simplify by adding and subtracting.

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Step 4.1.2.2.1

Add $- 1$ and $10$ .

$9 - 2$

Step 4.1.2.2.2

Subtract $2$ from $9$ .

$7$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

${((0) + 1)}^{5} - 5 \cdot 0 - 2$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Add $0$ and $1$ .

$1^{5} - 5 \cdot 0 - 2$

Step 4.2.2.1.2

One to any power is one.

$1 - 5 \cdot 0 - 2$

Step 4.2.2.1.3

Multiply $- 5$ by $0$ .

$1 + 0 - 2$

Step 4.2.2.2

Simplify by adding and subtracting.

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Step 4.2.2.2.1

Add $1$ and $0$ .

$1 - 2$

Step 4.2.2.2.2

Subtract $2$ from $1$ .

$- 1$

Step 4.3

List all of the points.

$(- 2, 7), (0, - 1)$

Step 5