Calculus Examples

$f (x) = 8 x^{3} + 7 x$

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

By the Sum Rule, the derivative of $8 x^{3} + 7 x$ with respect to $x$ is $\frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [7 x]$ .

$\frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [7 x]$

Step 1.1.2

Evaluate $\frac{d}{d x} [8 x^{3}]$ .

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Step 1.1.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{3}$ with respect to $x$ is $8 \frac{d}{d x} [x^{3}]$ .

$8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [7 x]$

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 $n = 3$ .

$8 (3 x^{2}) + \frac{d}{d x} [7 x]$

Step 1.1.2.3

Multiply $3$ by $8$ .

$24 x^{2} + \frac{d}{d x} [7 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [7 x]$ .

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Step 1.1.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$24 x^{2} + 7 \frac{d}{d x} [x]$

Step 1.1.3.2

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

$24 x^{2} + 7 \cdot 1$

Step 1.1.3.3

Multiply $7$ by $1$ .

$f' (x) = 24 x^{2} + 7$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $24 x^{2} + 7$ .

$24 x^{2} + 7$

Step 2

Set the first derivative equal to $0$ then solve the equation $24 x^{2} + 7 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$24 x^{2} + 7 = 0$

Step 2.2

Subtract $7$ from both sides of the equation.

$24 x^{2} = - 7$

Step 2.3

Divide each term in $24 x^{2} = - 7$ by $24$ and simplify.

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Step 2.3.1

Divide each term in $24 x^{2} = - 7$ by $24$ .

$\frac{24 x^{2}}{24} = \frac{- 7}{24}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $24$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{24 x^{2}}{24} = \frac{- 7}{24}$

Step 2.3.2.1.2

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

$x^{2} = \frac{- 7}{24}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{7}{24}$

Step 2.4

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

$x = \pm \sqrt{- \frac{7}{24}}$

Step 2.5

Simplify $\pm \sqrt{- \frac{7}{24}}$ .

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Step 2.5.1

Rewrite $- \frac{7}{24}$ as ${(\frac{i}{2})}^{2} \frac{7}{6}$ .

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Step 2.5.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{7}{24}}$

Step 2.5.1.2

Factor the perfect power $1^{2}$ out of $7$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 7}{24}}$

Step 2.5.1.3

Factor the perfect power $2^{2}$ out of $24$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 7}{2^{2} \cdot 6}}$

Step 2.5.1.4

Rearrange the fraction $\frac{1^{2} \cdot 7}{2^{2} \cdot 6}$ .

$x = \pm \sqrt{i^{2} ({(\frac{1}{2})}^{2} \frac{7}{6})}$

Step 2.5.1.5

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

$x = \pm \sqrt{{(\frac{i}{2})}^{2} \frac{7}{6}}$

Step 2.5.2

Pull terms out from under the radical.

$x = \pm \frac{i}{2} \sqrt{\frac{7}{6}}$

Step 2.5.3

Rewrite $\sqrt{\frac{7}{6}}$ as $\frac{\sqrt{7}}{\sqrt{6}}$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7}}{\sqrt{6}}$

Step 2.5.4

Multiply $\frac{\sqrt{7}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{i}{2} (\frac{\sqrt{7}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Step 2.5.5

Combine and simplify the denominator.

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Step 2.5.5.1

Multiply $\frac{\sqrt{7}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 2.5.5.2

Raise $\sqrt{6}$ to the power of $1$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 2.5.5.3

Raise $\sqrt{6}$ to the power of $1$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 2.5.5.4

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

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 2.5.5.5

Add $1$ and $1$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 2.5.5.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.5.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 2.5.5.6.2

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

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 2.5.5.6.3

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

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 2.5.5.6.4

Cancel the common factor of $2$ .

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Step 2.5.5.6.4.1

Cancel the common factor.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 2.5.5.6.4.2

Rewrite the expression.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7} \sqrt{6}}{6^{1}}$

Step 2.5.5.6.5

Evaluate the exponent.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7} \sqrt{6}}{6}$

Step 2.5.6

Simplify the numerator.

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Step 2.5.6.1

Combine using the product rule for radicals.

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{7 \cdot 6}}{6}$

Step 2.5.6.2

Multiply $7$ by $6$ .

$x = \pm \frac{i}{2} \cdot \frac{\sqrt{42}}{6}$

Step 2.5.7

Multiply $\frac{i}{2} \cdot \frac{\sqrt{42}}{6}$ .

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Step 2.5.7.1

Multiply $\frac{i}{2}$ by $\frac{\sqrt{42}}{6}$ .

$x = \pm \frac{i \sqrt{42}}{2 \cdot 6}$

Step 2.5.7.2

Multiply $2$ by $6$ .

$x = \pm \frac{i \sqrt{42}}{12}$

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.

$x = \frac{i \sqrt{42}}{12}$

Step 2.6.2

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

$x = - \frac{i \sqrt{42}}{12}$

Step 2.6.3

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

$x = \frac{i \sqrt{42}}{12}, - \frac{i \sqrt{42}}{12}$

Step 3

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 4

No points make the derivative $f' (x) = 24 x^{2} + 7$ equal to $0$ or undefined. The interval to check if $f (x) = 8 x^{3} + 7 x$ is increasing or decreasing is $(- \infty, \infty)$ .

$(- \infty, \infty)$

Step 5

Substitute any number, such as $1$ , from the interval $(- \infty, \infty)$ in the derivative $f' (x) = 24 x^{2} + 7$ 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 5.1

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

$f' (1) = 24 {(1)}^{2} + 7$

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

One to any power is one.

$f' (1) = 24 \cdot 1 + 7$

Step 5.2.1.2

Multiply $24$ by $1$ .

$f' (1) = 24 + 7$

Step 5.2.2

Add $24$ and $7$ .

$f' (1) = 31$

Step 5.2.3

The final answer is $31$ .

$31$

Step 6

The result of substituting $1$ into $f' (x) = 24 x^{2} + 7$ is $31$ , which is positive, so the graph is increasing on the interval $(- \infty, \infty)$ .

Increasing on $(- \infty, \infty)$ since $24 x^{2} + 7 > 0$

Step 7

Increasing over the interval $(- \infty, \infty)$ means that the function is always increasing.

Always Increasing

Step 8