Calculus Examples

$y = x^{3} - 3 x + 1$

Step 1

Set $y$ as a function of $x$ .

$f (x) = x^{3} - 3 x + 1$

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{3} - 3 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1]$ .

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $- 3$ by $1$ .

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

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

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

$3 x^{2} - 3 + 0$

Step 2.3.2

Add $3 x^{2} - 3$ and $0$ .

$3 x^{2} - 3$

Step 3

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

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Step 3.1

Add $3$ to both sides of the equation.

$3 x^{2} = 3$

Step 3.2

Divide each term in $3 x^{2} = 3$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 x^{2} = 3$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{3}{3}$

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.

$\frac{3 x^{2}}{3} = \frac{3}{3}$

Step 3.2.2.1.2

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

$x^{2} = \frac{3}{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $3$ by $3$ .

$x^{2} = 1$

Step 3.3

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

$x = \pm \sqrt{1}$

Step 3.4

Any root of $1$ is $1$ .

$x = \pm 1$

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.

$x = 1$

Step 3.5.2

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

$x = - 1$

Step 3.5.3

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

$x = 1, - 1$

Step 4

Solve the original function $f (x) = x^{3} - 3 x + 1$ at $x = 1$ .

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Step 4.1

Replace the variable $x$ with $1$ in the expression.

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

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

One to any power is one.

$f (1) = 1 - 3 \cdot 1 + 1$

Step 4.2.1.2

Multiply $- 3$ by $1$ .

$f (1) = 1 - 3 + 1$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $3$ from $1$ .

$f (1) = - 2 + 1$

Step 4.2.2.2

Add $- 2$ and $1$ .

$f (1) = - 1$

Step 4.2.3

The final answer is $- 1$ .

$- 1$

Step 5

Solve the original function $f (x) = x^{3} - 3 x + 1$ at $x = - 1$ .

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Step 5.1

Replace the variable $x$ with $- 1$ in the expression.

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

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

$f (- 1) = - 1 - 3 \cdot - 1 + 1$

Step 5.2.1.2

Multiply $- 3$ by $- 1$ .

$f (- 1) = - 1 + 3 + 1$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $- 1$ and $3$ .

$f (- 1) = 2 + 1$

Step 5.2.2.2

Add $2$ and $1$ .

$f (- 1) = 3$

Step 5.2.3

The final answer is $3$ .

$3$

Step 6

The horizontal tangent lines on function $f (x) = x^{3} - 3 x + 1$ are $y = - 1, y = 3$ .

$y = - 1, y = 3$

Step 7