Calculus Examples

y = 2 - 2 x - x^{3}

$y = 2 - 2 x - x^{3}$

Step 1

Write

$y = 2 - 2 x - x^{3}$ as a function.

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

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.

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Step 2.1.1.1

By the Sum Rule, the derivative of

$2 - 2 x - x^{3}$ with respect to

$x$ is

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

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

Step 2.1.1.2

Since

$2$ is constant with respect to

$x$ , the derivative of

$2$ with respect to

$x$ is

$0$ .

$0 + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- x^{3}]$

Step 2.1.2

Evaluate

$\frac{d}{d x} [- 2 x]$ .

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Step 2.1.2.1

Since

$- 2$ is constant with respect to

$x$ , the derivative of

$- 2 x$ with respect to

$x$ is

$- 2 \frac{d}{d x} [x]$ .

$0 - 2 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{3}]$

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

$n = 1$ .

$0 - 2 \cdot 1 + \frac{d}{d x} [- x^{3}]$

Step 2.1.2.3

Multiply

$- 2$ by

$1$ .

$0 - 2 + \frac{d}{d x} [- x^{3}]$

Step 2.1.3

Evaluate

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

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Step 2.1.3.1

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- x^{3}$ with respect to

$x$ is

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

$0 - 2 - \frac{d}{d x} [x^{3}]$

Step 2.1.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 3$ .

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

Step 2.1.3.3

Multiply

$3$ by

$- 1$ .

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

Step 2.1.4

Simplify.

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Step 2.1.4.1

Subtract

$2$ from

$0$ .

$- 2 - 3 x^{2}$

Step 2.1.4.2

Reorder terms.

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

Step 2.2

The first derivative of

$f (x)$ with respect to

$x$ is

$- 3 x^{2} - 2$ .

$- 3 x^{2} - 2$

Step 3

Set the first derivative equal to

$0$ then solve the equation

$- 3 x^{2} - 2 = 0$ .

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Step 3.1

Set the first derivative equal to

$0$ .

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

Step 3.2

Add

$2$ to both sides of the equation.

$- 3 x^{2} = 2$

Step 3.3

Divide each term in

$- 3 x^{2} = 2$ by

$- 3$ and simplify.

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Step 3.3.1

Divide each term in

$- 3 x^{2} = 2$ by

$- 3$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of

$- 3$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide

$x^{2}$ by

$1$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

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

Step 3.4

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

$x = \pm \sqrt{- \frac{2}{3}}$

Step 3.5

Simplify

$\pm \sqrt{- \frac{2}{3}}$ .

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Step 3.5.1

Rewrite

$- 1$ as

$i^{2}$ .

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

Step 3.5.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{2}{3}}$

Step 3.5.3

Rewrite

$\sqrt{\frac{2}{3}}$ as

$\frac{\sqrt{2}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{2}}{\sqrt{3}}$

Step 3.5.4

Multiply

$\frac{\sqrt{2}}{\sqrt{3}}$ by

$\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 3.5.5

Combine and simplify the denominator.

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Step 3.5.5.1

Multiply

$\frac{\sqrt{2}}{\sqrt{3}}$ by

$\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.5.5.2

Raise

$\sqrt{3}$ to the power of

$1$ .

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

Step 3.5.5.3

Raise

$\sqrt{3}$ to the power of

$1$ .

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

Step 3.5.5.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.5.5

Add

$1$ and

$1$ .

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

Step 3.5.5.6

Rewrite

${\sqrt{3}}^{2}$ as

$3$ .

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Step 3.5.5.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{3}$ as

$3^{\frac{1}{2}}$ .

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

Step 3.5.5.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.5.5.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 3.5.5.6.4

Cancel the common factor of

$2$ .

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Step 3.5.5.6.4.1

Cancel the common factor.

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

Step 3.5.5.6.4.2

Rewrite the expression.

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

Step 3.5.5.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{3}$

Step 3.5.6

Simplify the numerator.

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Step 3.5.6.1

Combine using the product rule for radicals.

$x = \pm i \frac{\sqrt{2 \cdot 3}}{3}$

Step 3.5.6.2

Multiply

$2$ by

$3$ .

$x = \pm i \frac{\sqrt{6}}{3}$

Step 3.5.7

Combine

$i$ and

$\frac{\sqrt{6}}{3}$ .

$x = \pm \frac{i \sqrt{6}}{3}$

Step 3.6

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

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Step 3.6.1

First, use the positive value of the

$\pm$ to find the first solution.

$x = \frac{i \sqrt{6}}{3}$

Step 3.6.2

Next, use the negative value of the

$\pm$ to find the second solution.

$x = - \frac{i \sqrt{6}}{3}$

Step 3.6.3

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

$x = \frac{i \sqrt{6}}{3}, - \frac{i \sqrt{6}}{3}$

Step 4

There are no values of

$x$ in the domain of the original problem where the derivative is

$0$ or undefined.

No critical points found

Step 5

No points make the derivative

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

$0$ or undefined. The interval to check if

$f (x) = 2 - 2 x - x^{3}$ is increasing or decreasing is

$(- \infty, \infty)$ .

$(- \infty, \infty)$

Step 6

Substitute any number, such as

$1$ , from the interval

$(- \infty, \infty)$ in the derivative

$f' (x) = - 3 x^{2} - 2$ to check if the result is negative or positive. If the result is negative, the graph is decreasing on the interval

$(- \infty, \infty)$ . If the result is positive, the graph is increasing on the interval

$(- \infty, \infty)$ .

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Step 6.1

Replace the variable

$x$ with

$1$ in the expression.

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

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

One to any power is one.

$f' (1) = - 3 \cdot 1 - 2$

Step 6.2.1.2

Multiply

$- 3$ by

$1$ .

$f' (1) = - 3 - 2$

Step 6.2.2

Subtract

$2$ from

$- 3$ .

$f' (1) = - 5$

Step 6.2.3

The final answer is

$- 5$ .

$- 5$

Step 7

The result of substituting

$1$ into

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

$- 5$ , which is negative, so the graph is decreasing on the interval

$(- \infty, \infty)$ .

Decreasing on

$(- \infty, \infty)$

Step 8

Decreasing over the interval

$(- \infty, \infty)$ means that the function is always decreasing.

Always Decreasing

Step 9