Calculus Examples

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

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

By the Sum Rule, the derivative of ${(x + 2)}^{7} - 7 x - 1$ with respect to $x$ is $\frac{d}{d x} [{(x + 2)}^{7}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 1]$ .

$f' (x) = \frac{d}{d x} ({(x + 2)}^{7}) + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (- 1)$

Step 1.1.2

Evaluate $\frac{d}{d x} [{(x + 2)}^{7}]$ .

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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 $f (x) = x^{7}$ and $g (x) = x + 2$ .

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Step 1.1.2.1.1

To apply the Chain Rule, set $u$ as $x + 2$ .

$f' (x) = \frac{d}{d u} (u^{7}) \frac{d}{d x} (x + 2) + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (- 1)$

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 $n = 7$ .

$f' (x) = 7 u^{6} \frac{d}{d x} (x + 2) + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (- 1)$

Step 1.1.2.1.3

Replace all occurrences of $u$ with $x + 2$ .

$f' (x) = 7 {(x + 2)}^{6} \frac{d}{d x} (x + 2) + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (- 1)$

Step 1.1.2.2

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

$f' (x) = 7 {(x + 2)}^{6} (\frac{d}{d x} (x) + \frac{d}{d x} (2)) + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (- 1)$

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 $n = 1$ .

$f' (x) = 7 {(x + 2)}^{6} (1 + \frac{d}{d x} (2)) + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (- 1)$

Step 1.1.2.4

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

$f' (x) = 7 {(x + 2)}^{6} (1 + 0) + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (- 1)$

Step 1.1.2.5

Add $1$ and $0$ .

$f' (x) = 7 {(x + 2)}^{6} \cdot 1 + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (- 1)$

Step 1.1.2.6

Multiply $7$ by $1$ .

$f' (x) = 7 {(x + 2)}^{6} + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (- 1)$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 7 x]$ .

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Step 1.1.3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$f' (x) = 7 {(x + 2)}^{6} - 7 \frac{d}{d x} x + \frac{d}{d x} (- 1)$

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 $n = 1$ .

$f' (x) = 7 {(x + 2)}^{6} - 7 \cdot 1 + \frac{d}{d x} (- 1)$

Step 1.1.3.3

Multiply $- 7$ by $1$ .

$f' (x) = 7 {(x + 2)}^{6} - 7 + \frac{d}{d x} (- 1)$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$f' (x) = 7 {(x + 2)}^{6} - 7 + 0$

Step 1.1.4.2

Add $7 {(x + 2)}^{6} - 7$ and $0$ .

$f' (x) = 7 {(x + 2)}^{6} - 7$

Step 1.2

The first derivative of $h (x)$ with respect to $x$ is $7 {(x + 2)}^{6} - 7$ .

$7 {(x + 2)}^{6} - 7$

Step 2

Set the first derivative equal to $0$ then solve the equation $7 {(x + 2)}^{6} - 7 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$7 {(x + 2)}^{6} - 7 = 0$

Step 2.2

Simplify $7 {(x + 2)}^{6} - 7$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Use the Binomial Theorem.

$7 (x^{6} + 6 x^{5} \cdot 2 + 15 x^{4} \cdot 2^{2} + 20 x^{3} \cdot 2^{3} + 15 x^{2} \cdot 2^{4} + 6 x \cdot 2^{5} + 2^{6}) - 7 = 0$

Step 2.2.1.2

Simplify each term.

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Step 2.2.1.2.1

Multiply $2$ by $6$ .

$7 (x^{6} + 12 x^{5} + 15 x^{4} \cdot 2^{2} + 20 x^{3} \cdot 2^{3} + 15 x^{2} \cdot 2^{4} + 6 x \cdot 2^{5} + 2^{6}) - 7 = 0$

Step 2.2.1.2.2

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

$7 (x^{6} + 12 x^{5} + 15 x^{4} \cdot 4 + 20 x^{3} \cdot 2^{3} + 15 x^{2} \cdot 2^{4} + 6 x \cdot 2^{5} + 2^{6}) - 7 = 0$

Step 2.2.1.2.3

Multiply $4$ by $15$ .

$7 (x^{6} + 12 x^{5} + 60 x^{4} + 20 x^{3} \cdot 2^{3} + 15 x^{2} \cdot 2^{4} + 6 x \cdot 2^{5} + 2^{6}) - 7 = 0$

Step 2.2.1.2.4

Raise $2$ to the power of $3$ .

$7 (x^{6} + 12 x^{5} + 60 x^{4} + 20 x^{3} \cdot 8 + 15 x^{2} \cdot 2^{4} + 6 x \cdot 2^{5} + 2^{6}) - 7 = 0$

Step 2.2.1.2.5

Multiply $8$ by $20$ .

$7 (x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 15 x^{2} \cdot 2^{4} + 6 x \cdot 2^{5} + 2^{6}) - 7 = 0$

Step 2.2.1.2.6

Raise $2$ to the power of $4$ .

$7 (x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 15 x^{2} \cdot 16 + 6 x \cdot 2^{5} + 2^{6}) - 7 = 0$

Step 2.2.1.2.7

Multiply $16$ by $15$ .

$7 (x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 6 x \cdot 2^{5} + 2^{6}) - 7 = 0$

Step 2.2.1.2.8

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

$7 (x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 6 x \cdot 32 + 2^{6}) - 7 = 0$

Step 2.2.1.2.9

Multiply $32$ by $6$ .

$7 (x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 2^{6}) - 7 = 0$

Step 2.2.1.2.10

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

$7 (x^{6} + 12 x^{5} + 60 x^{4} + 160 x^{3} + 240 x^{2} + 192 x + 64) - 7 = 0$

Step 2.2.1.3

Apply the distributive property.

$7 x^{6} + 7 (12 x^{5}) + 7 (60 x^{4}) + 7 (160 x^{3}) + 7 (240 x^{2}) + 7 (192 x) + 7 \cdot 64 - 7 = 0$

Step 2.2.1.4

Simplify.

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Step 2.2.1.4.1

Multiply $12$ by $7$ .

$7 x^{6} + 84 x^{5} + 7 (60 x^{4}) + 7 (160 x^{3}) + 7 (240 x^{2}) + 7 (192 x) + 7 \cdot 64 - 7 = 0$

Step 2.2.1.4.2

Multiply $60$ by $7$ .

$7 x^{6} + 84 x^{5} + 420 x^{4} + 7 (160 x^{3}) + 7 (240 x^{2}) + 7 (192 x) + 7 \cdot 64 - 7 = 0$

Step 2.2.1.4.3

Multiply $160$ by $7$ .

$7 x^{6} + 84 x^{5} + 420 x^{4} + 1120 x^{3} + 7 (240 x^{2}) + 7 (192 x) + 7 \cdot 64 - 7 = 0$

Step 2.2.1.4.4

Multiply $240$ by $7$ .

$7 x^{6} + 84 x^{5} + 420 x^{4} + 1120 x^{3} + 1680 x^{2} + 7 (192 x) + 7 \cdot 64 - 7 = 0$

Step 2.2.1.4.5

Multiply $192$ by $7$ .

$7 x^{6} + 84 x^{5} + 420 x^{4} + 1120 x^{3} + 1680 x^{2} + 1344 x + 7 \cdot 64 - 7 = 0$

Step 2.2.1.4.6

Multiply $7$ by $64$ .

$7 x^{6} + 84 x^{5} + 420 x^{4} + 1120 x^{3} + 1680 x^{2} + 1344 x + 448 - 7 = 0$

Step 2.2.2

Subtract $7$ from $448$ .

$7 x^{6} + 84 x^{5} + 420 x^{4} + 1120 x^{3} + 1680 x^{2} + 1344 x + 441 = 0$

Step 2.3

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

$x = - 3, - 1$

Step 3

The values which make the derivative equal to $0$ are $- 3, - 1$ .

$- 3, - 1$

Step 4

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

$(- \infty, - 3) \cup (- 3, - 1) \cup (- 1, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 4$ in the expression.

$h' (- 4) = 7 {((- 4) + 2)}^{6} - 7$

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

Add $- 4$ and $2$ .

$h' (- 4) = 7 {(- 2)}^{6} - 7$

Step 5.2.1.2

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

$h' (- 4) = 7 \cdot 64 - 7$

Step 5.2.1.3

Multiply $7$ by $64$ .

$h' (- 4) = 448 - 7$

Step 5.2.2

Subtract $7$ from $448$ .

$h' (- 4) = 441$

Step 5.2.3

The final answer is $441$ .

$441$

Step 5.3

At $x = - 4$ the derivative is $441$ . Since this is positive, the function is increasing on $(- \infty, - 3)$ .

Increasing on $(- \infty, - 3)$ since $h' (x) > 0$

Step 6

Substitute a value from the interval $(- 3, - 1)$ 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.

$h' (- 2) = 7 {((- 2) + 2)}^{6} - 7$

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

Add $- 2$ and $2$ .

$h' (- 2) = 7 \cdot 0^{6} - 7$

Step 6.2.1.2

Raising $0$ to any positive power yields $0$ .

$h' (- 2) = 7 \cdot 0 - 7$

Step 6.2.1.3

Multiply $7$ by $0$ .

$h' (- 2) = 0 - 7$

Step 6.2.2

Subtract $7$ from $0$ .

$h' (- 2) = - 7$

Step 6.2.3

The final answer is $- 7$ .

$- 7$

Step 6.3

At $x = - 2$ the derivative is $- 7$ . Since this is negative, the function is decreasing on $(- 3, - 1)$ .

Decreasing on $(- 3, - 1)$ since $h' (x) < 0$

Step 7

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

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Step 7.1

Replace the variable $x$ with $0$ in the expression.

$h' (0) = 7 {((0) + 2)}^{6} - 7$

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

Add $0$ and $2$ .

$h' (0) = 7 \cdot 2^{6} - 7$

Step 7.2.1.2

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

$h' (0) = 7 \cdot 64 - 7$

Step 7.2.1.3

Multiply $7$ by $64$ .

$h' (0) = 448 - 7$

Step 7.2.2

Subtract $7$ from $448$ .

$h' (0) = 441$

Step 7.2.3

The final answer is $441$ .

$441$

Step 7.3

At $x = 0$ the derivative is $441$ . Since this is positive, the function is increasing on $(- 1, \infty)$ .

Increasing on $(- 1, \infty)$ since $h' (x) > 0$

Step 8

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

Increasing on: $(- \infty, - 3), (- 1, \infty)$

Decreasing on: $(- 3, - 1)$

Step 9