Calculus Examples

$y = 3 x^{3} - x - 3$

Step 1

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

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

Step 2

Find the first derivative.

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Step 2.1

By the Sum Rule, the derivative of $3 x^{3} - x - 3$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 3]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [3 x^{3}]$ .

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Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{3}$ with respect to $x$ is $3 \frac{d}{d x} [x^{3}]$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $3$ by $3$ .

$9 x^{2} + \frac{d}{d x} [- x] + \frac{d}{d x} [- 3]$

Step 2.3

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

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Step 2.3.1

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

$9 x^{2} - \frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 2.3.2

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

$9 x^{2} - 1 \cdot 1 + \frac{d}{d x} [- 3]$

Step 2.3.3

Multiply $- 1$ by $1$ .

$9 x^{2} - 1 + \frac{d}{d x} [- 3]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$9 x^{2} - 1 + 0$

Step 2.4.2

Add $9 x^{2} - 1$ and $0$ .

$9 x^{2} - 1$

Step 3

Set the first derivative equal to $0$ and solve for $x$ .

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Step 3.1

Add $1$ to both sides of the equation.

$9 x^{2} = 1$

Step 3.2

Divide each term in $9 x^{2} = 1$ by $9$ and simplify.

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Step 3.2.1

Divide each term in $9 x^{2} = 1$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{1}{9}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $9$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{9 x^{2}}{9} = \frac{1}{9}$

Step 3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{1}{9}$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{9}}$

Step 3.4

Simplify $\pm \sqrt{\frac{1}{9}}$ .

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Step 3.4.1

Rewrite $\sqrt{\frac{1}{9}}$ as $\frac{\sqrt{1}}{\sqrt{9}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{9}}$

Step 3.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{9}}$

Step 3.4.3

Simplify the denominator.

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Step 3.4.3.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \frac{1}{\sqrt{3^{2}}}$

Step 3.4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{1}{3}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{1}{3}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{1}{3}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{1}{3}, - \frac{1}{3}$

Step 4

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

$(- \infty, - \frac{1}{3}) \cup (- \frac{1}{3}, \frac{1}{3}) \cup (\frac{1}{3}, \infty)$

Step 5

Substitute any number, such as $- 3$ , from the interval $(- \infty, - \frac{1}{3})$ in the first derivative $9 x^{2} - 1$ to check if the result is negative or positive.

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Step 5.1

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

$f' (- 3) = 9 {(- 3)}^{2} - 1$

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

$f' (- 3) = 9 \cdot 9 - 1$

Step 5.2.1.2

Multiply $9$ by $9$ .

$f' (- 3) = 81 - 1$

Step 5.2.2

Subtract $1$ from $81$ .

$f' (- 3) = 80$

Step 5.2.3

The final answer is $80$ .

$80$

Step 6

Substitute any number, such as $0$ , from the interval $(- \frac{1}{3}, \frac{1}{3})$ in the first derivative $9 x^{2} - 1$ to check if the result is negative or positive.

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Step 6.1

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

$f' (0) = 9 {(0)}^{2} - 1$

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

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

$f' (0) = 9 \cdot 0 - 1$

Step 6.2.1.2

Multiply $9$ by $0$ .

$f' (0) = 0 - 1$

Step 6.2.2

Subtract $1$ from $0$ .

$f' (0) = - 1$

Step 6.2.3

The final answer is $- 1$ .

$- 1$

Step 7

Substitute any number, such as $3$ , from the interval $(\frac{1}{3}, \infty)$ in the first derivative $9 x^{2} - 1$ to check if the result is negative or positive.

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Step 7.1

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

$f' (3) = 9 {(3)}^{2} - 1$

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 $3$ to the power of $2$ .

$f' (3) = 9 \cdot 9 - 1$

Step 7.2.1.2

Multiply $9$ by $9$ .

$f' (3) = 81 - 1$

Step 7.2.2

Subtract $1$ from $81$ .

$f' (3) = 80$

Step 7.2.3

The final answer is $80$ .

$80$

Step 8

Since the first derivative changed signs from positive to negative around $x = - \frac{1}{3}$ , then there is a turning point at $x = - \frac{1}{3}$ .

Step 9

Find the y-coordinate of $- \frac{1}{3}$ to find the turning point.

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Step 9.1

Find $f (- \frac{1}{3})$ to find the y-coordinate of $- \frac{1}{3}$ .

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Step 9.1.1

Replace the variable $x$ with $- \frac{1}{3}$ in the expression.

$f (- \frac{1}{3}) = 3 {(- \frac{1}{3})}^{3} - (- \frac{1}{3}) - 3$

Step 9.1.2

Simplify $3 {(- \frac{1}{3})}^{3} - (- \frac{1}{3}) - 3$ .

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Step 9.1.2.1

Remove parentheses.

$3 {(- \frac{1}{3})}^{3} - (- \frac{1}{3}) - 3$

Step 9.1.2.2

Simplify each term.

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Step 9.1.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.1.2.2.1.1

Apply the product rule to $- \frac{1}{3}$ .

$3 ({(- 1)}^{3} {(\frac{1}{3})}^{3}) - (- \frac{1}{3}) - 3$

Step 9.1.2.2.1.2

Apply the product rule to $\frac{1}{3}$ .

$3 ({(- 1)}^{3} \frac{1^{3}}{3^{3}}) - (- \frac{1}{3}) - 3$

Step 9.1.2.2.2

Raise $- 1$ to the power of $3$ .

$3 (- \frac{1^{3}}{3^{3}}) - (- \frac{1}{3}) - 3$

Step 9.1.2.2.3

One to any power is one.

$3 (- \frac{1}{3^{3}}) - (- \frac{1}{3}) - 3$

Step 9.1.2.2.4

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

$3 (- \frac{1}{27}) - (- \frac{1}{3}) - 3$

Step 9.1.2.2.5

Cancel the common factor of $3$ .

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Step 9.1.2.2.5.1

Move the leading negative in $- \frac{1}{27}$ into the numerator.

$3 (\frac{- 1}{27}) - (- \frac{1}{3}) - 3$

Step 9.1.2.2.5.2

Factor $3$ out of $27$ .

$3 \frac{- 1}{3 (9)} - (- \frac{1}{3}) - 3$

Step 9.1.2.2.5.3

Cancel the common factor.

$3 \frac{- 1}{3 \cdot 9} - (- \frac{1}{3}) - 3$

Step 9.1.2.2.5.4

Rewrite the expression.

$\frac{- 1}{9} - (- \frac{1}{3}) - 3$

Step 9.1.2.2.6

Move the negative in front of the fraction.

$- \frac{1}{9} - (- \frac{1}{3}) - 3$

Step 9.1.2.2.7

Multiply $- (- \frac{1}{3})$ .

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Step 9.1.2.2.7.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{9} + 1 (\frac{1}{3}) - 3$

Step 9.1.2.2.7.2

Multiply $\frac{1}{3}$ by $1$ .

$- \frac{1}{9} + \frac{1}{3} - 3$

Step 9.1.2.3

Find the common denominator.

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Step 9.1.2.3.1

Multiply $\frac{1}{3}$ by $\frac{3}{3}$ .

$- \frac{1}{9} + \frac{1}{3} \cdot \frac{3}{3} - 3$

Step 9.1.2.3.2

Multiply $\frac{1}{3}$ by $\frac{3}{3}$ .

$- \frac{1}{9} + \frac{3}{3 \cdot 3} - 3$

Step 9.1.2.3.3

Write $- 3$ as a fraction with denominator $1$ .

$- \frac{1}{9} + \frac{3}{3 \cdot 3} + \frac{- 3}{1}$

Step 9.1.2.3.4

Multiply $\frac{- 3}{1}$ by $\frac{9}{9}$ .

$- \frac{1}{9} + \frac{3}{3 \cdot 3} + \frac{- 3}{1} \cdot \frac{9}{9}$

Step 9.1.2.3.5

Multiply $\frac{- 3}{1}$ by $\frac{9}{9}$ .

$- \frac{1}{9} + \frac{3}{3 \cdot 3} + \frac{- 3 \cdot 9}{9}$

Step 9.1.2.3.6

Multiply $3$ by $3$ .

$- \frac{1}{9} + \frac{3}{9} + \frac{- 3 \cdot 9}{9}$

Step 9.1.2.4

Combine the numerators over the common denominator.

$\frac{- 1 + 3 - 3 \cdot 9}{9}$

Step 9.1.2.5

Simplify the expression.

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Step 9.1.2.5.1

Multiply $- 3$ by $9$ .

$\frac{- 1 + 3 - 27}{9}$

Step 9.1.2.5.2

Add $- 1$ and $3$ .

$\frac{2 - 27}{9}$

Step 9.1.2.5.3

Subtract $27$ from $2$ .

$\frac{- 25}{9}$

Step 9.1.2.5.4

Move the negative in front of the fraction.

$- \frac{25}{9}$

Step 9.2

Write the $x$ and $y$ coordinates in point form.

$(- \frac{1}{3}, - \frac{25}{9})$

Step 10

Since the first derivative changed signs from negative to positive around $x = \frac{1}{3}$ , then there is a turning point at $x = \frac{1}{3}$ .

Step 11

Find the y-coordinate of $\frac{1}{3}$ to find the turning point.

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Step 11.1

Find $f (\frac{1}{3})$ to find the y-coordinate of $\frac{1}{3}$ .

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Step 11.1.1

Replace the variable $x$ with $\frac{1}{3}$ in the expression.

$f (\frac{1}{3}) = 3 {(\frac{1}{3})}^{3} - (\frac{1}{3}) - 3$

Step 11.1.2

Simplify $3 {(\frac{1}{3})}^{3} - (\frac{1}{3}) - 3$ .

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Step 11.1.2.1

Remove parentheses.

$3 {(\frac{1}{3})}^{3} - (\frac{1}{3}) - 3$

Step 11.1.2.2

Simplify each term.

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Step 11.1.2.2.1

Apply the product rule to $\frac{1}{3}$ .

$3 \frac{1^{3}}{3^{3}} - (\frac{1}{3}) - 3$

Step 11.1.2.2.2

One to any power is one.

$3 \frac{1}{3^{3}} - (\frac{1}{3}) - 3$

Step 11.1.2.2.3

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

$3 (\frac{1}{27}) - (\frac{1}{3}) - 3$

Step 11.1.2.2.4

Cancel the common factor of $3$ .

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Step 11.1.2.2.4.1

Factor $3$ out of $27$ .

$3 \frac{1}{3 (9)} - (\frac{1}{3}) - 3$

Step 11.1.2.2.4.2

Cancel the common factor.

$3 \frac{1}{3 \cdot 9} - (\frac{1}{3}) - 3$

Step 11.1.2.2.4.3

Rewrite the expression.

$\frac{1}{9} - (\frac{1}{3}) - 3$

$\frac{1}{9} - \frac{1}{3} - 3$

Step 11.1.2.3

Find the common denominator.

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Step 11.1.2.3.1

Multiply $\frac{1}{3}$ by $\frac{3}{3}$ .

$\frac{1}{9} - (\frac{1}{3} \cdot \frac{3}{3}) - 3$

Step 11.1.2.3.2

Multiply $\frac{1}{3}$ by $\frac{3}{3}$ .

$\frac{1}{9} - \frac{3}{3 \cdot 3} - 3$

Step 11.1.2.3.3

Write $- 3$ as a fraction with denominator $1$ .

$\frac{1}{9} - \frac{3}{3 \cdot 3} + \frac{- 3}{1}$

Step 11.1.2.3.4

Multiply $\frac{- 3}{1}$ by $\frac{9}{9}$ .

$\frac{1}{9} - \frac{3}{3 \cdot 3} + \frac{- 3}{1} \cdot \frac{9}{9}$

Step 11.1.2.3.5

Multiply $\frac{- 3}{1}$ by $\frac{9}{9}$ .

$\frac{1}{9} - \frac{3}{3 \cdot 3} + \frac{- 3 \cdot 9}{9}$

Step 11.1.2.3.6

Multiply $3$ by $3$ .

$\frac{1}{9} - \frac{3}{9} + \frac{- 3 \cdot 9}{9}$

Step 11.1.2.4

Combine the numerators over the common denominator.

$\frac{1 - 3 - 3 \cdot 9}{9}$

Step 11.1.2.5

Simplify the expression.

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Step 11.1.2.5.1

Multiply $- 3$ by $9$ .

$\frac{1 - 3 - 27}{9}$

Step 11.1.2.5.2

Subtract $3$ from $1$ .

$\frac{- 2 - 27}{9}$

Step 11.1.2.5.3

Subtract $27$ from $- 2$ .

$\frac{- 29}{9}$

Step 11.1.2.5.4

Move the negative in front of the fraction.

$- \frac{29}{9}$

Step 11.2

Write the $x$ and $y$ coordinates in point form.

$(\frac{1}{3}, - \frac{29}{9})$

Step 12

These are the turning points.

$(- \frac{1}{3}, - \frac{25}{9})$

$(\frac{1}{3}, - \frac{29}{9})$

Step 13