Calculus Examples

$y = {(x - 4)}^{3}$

Step 1

Write $y = {(x - 4)}^{3}$ as a function.

$f (x) = {(x - 4)}^{3}$

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x - 4$ .

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Step 2.1.1.1

To apply the Chain Rule, set $u$ as $x - 4$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [x - 4]$

Step 2.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 u^{2} \frac{d}{d x} [x - 4]$

Step 2.1.1.3

Replace all occurrences of $u$ with $x - 4$ .

$3 {(x - 4)}^{2} \frac{d}{d x} [x - 4]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

$3 {(x - 4)}^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 4])$

Step 2.1.2.2

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

$3 {(x - 4)}^{2} (1 + \frac{d}{d x} [- 4])$

Step 2.1.2.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

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

Step 2.1.2.4

Simplify the expression.

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Step 2.1.2.4.1

Add $1$ and $0$ .

$3 {(x - 4)}^{2} \cdot 1$

Step 2.1.2.4.2

Multiply $3$ by $1$ .

$f' (x) = 3 {(x - 4)}^{2}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $3 {(x - 4)}^{2}$ .

$3 {(x - 4)}^{2}$

Step 3

Set the first derivative equal to $0$ then solve the equation $3 {(x - 4)}^{2} = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

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

Step 3.2

Divide each term in $3 {(x - 4)}^{2} = 0$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 {(x - 4)}^{2} = 0$ by $3$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide ${(x - 4)}^{2}$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $0$ by $3$ .

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

Step 3.3

Set the $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 3.4

Add $4$ to both sides of the equation.

$x = 4$

Step 4

The values which make the derivative equal to $0$ are $4$ .

$4$

Step 5

After finding the point that makes the derivative $f' (x) = 3 {(x - 4)}^{2}$ equal to $0$ or undefined, the interval to check where $f (x) = {(x - 4)}^{3}$ is increasing and where it is decreasing is $(- \infty, 4) \cup (4, \infty)$ .

$(- \infty, 4) \cup (4, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 4)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = 3 {((3) - 4)}^{2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Subtract $4$ from $3$ .

$f' (3) = 3 {(- 1)}^{2}$

Step 6.2.2

Raise $- 1$ to the power of $2$ .

$f' (3) = 3 \cdot 1$

Step 6.2.3

Multiply $3$ by $1$ .

$f' (3) = 3$

Step 6.2.4

The final answer is $3$ .

$3$

Step 6.3

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

Increasing on $(- \infty, 4)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(4, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

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

$f' (5) = 3 {((5) - 4)}^{2}$

Step 7.2

Simplify the result.

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Step 7.2.1

Subtract $4$ from $5$ .

$f' (5) = 3 \cdot 1^{2}$

Step 7.2.2

One to any power is one.

$f' (5) = 3 \cdot 1$

Step 7.2.3

Multiply $3$ by $1$ .

$f' (5) = 3$

Step 7.2.4

The final answer is $3$ .

$3$

Step 7.3

At $x = 5$ the derivative is $3$ . Since this is positive, the function is increasing on $(4, \infty)$ .

Increasing on $(4, \infty)$ since $f' (x) > 0$

Step 8

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

Increasing on: $(- \infty, 4), (4, \infty)$

Step 9