Calculus Examples

$f (x) = 8 x^{3} + x^{2} + 8 x$

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 $8 x^{3} + x^{2} + 8 x$ with respect to $x$ is $\frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x]$ .

$f' (x) = \frac{d}{d x} (8 x^{3}) + \frac{d}{d x} (x^{2}) + \frac{d}{d x} (8 x)$

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

$f' (x) = 8 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (x^{2}) + \frac{d}{d x} (8 x)$

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 $n = 3$ .

$f' (x) = 8 (3 x^{2}) + \frac{d}{d x} (x^{2}) + \frac{d}{d x} (8 x)$

Step 1.1.2.3

Multiply $3$ by $8$ .

$f' (x) = 24 x^{2} + \frac{d}{d x} (x^{2}) + \frac{d}{d x} (8 x)$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = 24 x^{2} + 2 x + \frac{d}{d x} (8 x)$

Step 1.1.4

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

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Step 1.1.4.1

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

$f' (x) = 24 x^{2} + 2 x + 8 \frac{d}{d x} (x)$

Step 1.1.4.2

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

$f' (x) = 24 x^{2} + 2 x + 8 \cdot 1$

Step 1.1.4.3

Multiply $8$ by $1$ .

$f' (x) = 24 x^{2} + 2 x + 8$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $24 x^{2} + 2 x + 8$ .

$24 x^{2} + 2 x + 8$

Step 2

Set the first derivative equal to $0$ then solve the equation $24 x^{2} + 2 x + 8 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$24 x^{2} + 2 x + 8 = 0$

Step 2.2

Factor $2$ out of $24 x^{2} + 2 x + 8$ .

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Step 2.2.1

Factor $2$ out of $24 x^{2}$ .

$2 (12 x^{2}) + 2 x + 8 = 0$

Step 2.2.2

Factor $2$ out of $2 x$ .

$2 (12 x^{2}) + 2 (x) + 8 = 0$

Step 2.2.3

Factor $2$ out of $8$ .

$2 (12 x^{2}) + 2 x + 2 \cdot 4 = 0$

Step 2.2.4

Factor $2$ out of $2 (12 x^{2}) + 2 x$ .

$2 (12 x^{2} + x) + 2 \cdot 4 = 0$

Step 2.2.5

Factor $2$ out of $2 (12 x^{2} + x) + 2 \cdot 4$ .

$2 (12 x^{2} + x + 4) = 0$

Step 2.3

Divide each term in $2 (12 x^{2} + x + 4) = 0$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 (12 x^{2} + x + 4) = 0$ by $2$ .

$\frac{2 (12 x^{2} + x + 4)}{2} = \frac{0}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 (12 x^{2} + x + 4)}{2} = \frac{0}{2}$

Step 2.3.2.1.2

Divide $12 x^{2} + x + 4$ by $1$ .

$12 x^{2} + x + 4 = \frac{0}{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $2$ .

$12 x^{2} + x + 4 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.5

Substitute the values $a = 12$ , $b = 1$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (12 \cdot 4)}}{2 \cdot 12}$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 12 \cdot 4}}{2 \cdot 12}$

Step 2.6.1.2

Multiply $- 4 \cdot 12 \cdot 4$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{- 1 \pm \sqrt{1 - 48 \cdot 4}}{2 \cdot 12}$

Step 2.6.1.2.2

Multiply $- 48$ by $4$ .

$x = \frac{- 1 \pm \sqrt{1 - 192}}{2 \cdot 12}$

Step 2.6.1.3

Subtract $192$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 191}}{2 \cdot 12}$

Step 2.6.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 191}}{2 \cdot 12}$

Step 2.6.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot 12}$

Step 2.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{191}}{2 \cdot 12}$

Step 2.6.2

Multiply $2$ by $12$ .

$x = \frac{- 1 \pm i \sqrt{191}}{24}$

Step 2.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 12 \cdot 4}}{2 \cdot 12}$

Step 2.7.1.2

Multiply $- 4 \cdot 12 \cdot 4$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{- 1 \pm \sqrt{1 - 48 \cdot 4}}{2 \cdot 12}$

Step 2.7.1.2.2

Multiply $- 48$ by $4$ .

$x = \frac{- 1 \pm \sqrt{1 - 192}}{2 \cdot 12}$

Step 2.7.1.3

Subtract $192$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 191}}{2 \cdot 12}$

Step 2.7.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 191}}{2 \cdot 12}$

Step 2.7.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot 12}$

Step 2.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{191}}{2 \cdot 12}$

Step 2.7.2

Multiply $2$ by $12$ .

$x = \frac{- 1 \pm i \sqrt{191}}{24}$

Step 2.7.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{191}}{24}$

Step 2.7.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + i \sqrt{191}}{24}$

Step 2.7.5

Factor $- 1$ out of $i \sqrt{191}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{191})}{24}$

Step 2.7.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{191})$ .

$x = \frac{- 1 (1 - i \sqrt{191})}{24}$

Step 2.7.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{191}}{24}$

Step 2.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 12 \cdot 4}}{2 \cdot 12}$

Step 2.8.1.2

Multiply $- 4 \cdot 12 \cdot 4$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{- 1 \pm \sqrt{1 - 48 \cdot 4}}{2 \cdot 12}$

Step 2.8.1.2.2

Multiply $- 48$ by $4$ .

$x = \frac{- 1 \pm \sqrt{1 - 192}}{2 \cdot 12}$

Step 2.8.1.3

Subtract $192$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 191}}{2 \cdot 12}$

Step 2.8.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 191}}{2 \cdot 12}$

Step 2.8.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot 12}$

Step 2.8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{191}}{2 \cdot 12}$

Step 2.8.2

Multiply $2$ by $12$ .

$x = \frac{- 1 \pm i \sqrt{191}}{24}$

Step 2.8.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{191}}{24}$

Step 2.8.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - i \sqrt{191}}{24}$

Step 2.8.5

Factor $- 1$ out of $- i \sqrt{191}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{191})}{24}$

Step 2.8.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{191})$ .

$x = \frac{- 1 (1 + i \sqrt{191})}{24}$

Step 2.8.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{191}}{24}$

Step 2.9

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{191}}{24}, - \frac{1 + i \sqrt{191}}{24}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

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