Calculus Examples

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

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

Differentiate.

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Step 1.1.1.1

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

$f' (x) = \frac{d}{d x} (x) + \frac{d}{d x} (- x^{3})$

Step 1.1.1.2

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

$f' (x) = 1 + \frac{d}{d x} (- x^{3})$

Step 1.1.2

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

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Step 1.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Multiply $3$ by $- 1$ .

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

Step 1.1.3

Reorder terms.

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 3 x^{2} + 1$ .

$- 3 x^{2} + 1$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 3 x^{2} + 1 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 3 x^{2} + 1 = 0$

Step 2.2

Subtract $1$ from both sides of the equation.

$- 3 x^{2} = - 1$

Step 2.3

Divide each term in $- 3 x^{2} = - 1$ by $- 3$ and simplify.

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Step 2.3.1

Divide each term in $- 3 x^{2} = - 1$ by $- 3$ .

$\frac{- 3 x^{2}}{- 3} = \frac{- 1}{- 3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 3 x^{2}}{- 3} = \frac{- 1}{- 3}$

Step 2.3.2.1.2

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

$x^{2} = \frac{- 1}{- 3}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Dividing two negative values results in a positive value.

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

Step 2.4

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

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

Step 2.5

Simplify $\pm \sqrt{\frac{1}{3}}$ .

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Step 2.5.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

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

Step 2.5.2

Any root of $1$ is $1$ .

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

Step 2.5.3

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

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

Step 2.5.4

Combine and simplify the denominator.

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Step 2.5.4.1

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

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

Step 2.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 2.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 2.5.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.5.4.5

Add $1$ and $1$ .

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

Step 2.5.4.6

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

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Step 2.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 2.5.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.5.4.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 2.5.4.6.4

Cancel the common factor of $2$ .

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Step 2.5.4.6.4.1

Cancel the common factor.

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

Step 2.5.4.6.4.2

Rewrite the expression.

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

Step 2.5.4.6.5

Evaluate the exponent.

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 3

The values which make the derivative equal to $0$ are $\frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$ .

$\frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Step 4

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

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

Step 5

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

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Step 5.1

Replace the variable $x$ with $- 1.57735026$ in the expression.

$f' (- 1.57735026) = - 3 {(- 1.57735026)}^{2} + 1$

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 $- 1.57735026$ to the power of $2$ .

$f' (- 1.57735026) = - 3 \cdot 2.48803384 + 1$

Step 5.2.1.2

Multiply $- 3$ by $2.48803384$ .

$f' (- 1.57735026) = - 7.46410152 + 1$

Step 5.2.2

Add $- 7.46410152$ and $1$ .

$f' (- 1.57735026) = - 6.46410152$

Step 5.2.3

The final answer is $- 6.46410152$ .

$- 6.46410152$

Step 5.3

At $x = - 1.57735026$ the derivative is $- 6.46410152$ . Since this is negative, the function is decreasing on $(- \infty, - \frac{\sqrt{3}}{3})$ .

Decreasing on $(- \infty, - \frac{\sqrt{3}}{3})$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(- 0.57735026, \frac{\sqrt{3}}{3})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

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

$f' (0) = - 3 {(0)}^{2} + 1$

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 $0$ .

$f' (0) = - 3 \cdot 0 + 1$

Step 6.2.1.2

Multiply $- 3$ by $0$ .

$f' (0) = 0 + 1$

Step 6.2.2

Add $0$ and $1$ .

$f' (0) = 1$

Step 6.2.3

The final answer is $1$ .

$1$

Step 6.3

At $x = 0$ the derivative is $1$ . Since this is positive, the function is increasing on $(- 0.57735026, \frac{\sqrt{3}}{3})$ .

Increasing on $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ since $f' (x) > 0$

Step 7

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

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Step 7.1

Replace the variable $x$ with $1.57735026$ in the expression.

$f' (1.57735026) = - 3 {(1.57735026)}^{2} + 1$

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 $1.57735026$ to the power of $2$ .

$f' (1.57735026) = - 3 \cdot 2.48803384 + 1$

Step 7.2.1.2

Multiply $- 3$ by $2.48803384$ .

$f' (1.57735026) = - 7.46410152 + 1$

Step 7.2.2

Add $- 7.46410152$ and $1$ .

$f' (1.57735026) = - 6.46410152$

Step 7.2.3

The final answer is $- 6.46410152$ .

$- 6.46410152$

Step 7.3

At $x = 1.57735026$ the derivative is $- 6.46410152$ . Since this is negative, the function is decreasing on $(\frac{\sqrt{3}}{3}, \infty)$ .

Decreasing on $(\frac{\sqrt{3}}{3}, \infty)$ since $f' (x) < 0$

Step 8

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

Increasing on: $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$

Decreasing on: $(- \infty, - \frac{\sqrt{3}}{3}), (\frac{\sqrt{3}}{3}, \infty)$

Step 9