Calculus Examples

$f (x) = 4 x^{3} + x^{2} + 4 x$

Step 1

Find the first derivative.

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Find the first derivative.

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By the Sum Rule, the derivative of $4 x^{3} + x^{2} + 4 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x]$ .

$f' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (x^{2}) + \frac{d}{d x} (4 x)$

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (x^{2}) + \frac{d}{d x} (4 x)$

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

$f' (x) = 4 (3 x^{2}) + \frac{d}{d x} (x^{2}) + \frac{d}{d x} (4 x)$

Multiply $3$ by $4$ .

$f' (x) = 12 x^{2} + \frac{d}{d x} (x^{2}) + \frac{d}{d x} (4 x)$

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

$f' (x) = 12 x^{2} + 2 x + \frac{d}{d x} (4 x)$

Evaluate $\frac{d}{d x} [4 x]$ .

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Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$f' (x) = 12 x^{2} + 2 x + 4 \frac{d}{d x} (x)$

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

$f' (x) = 12 x^{2} + 2 x + 4 \cdot 1$

Multiply $4$ by $1$ .

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

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} + 2 x + 4$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $12 x^{2} + 2 x + 4 = 0$ .

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Set the first derivative equal to $0$ .

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

Factor $2$ out of $12 x^{2} + 2 x + 4$ .

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Factor $2$ out of $12 x^{2}$ .

$2 (6 x^{2}) + 2 x + 4 = 0$

Factor $2$ out of $2 x$ .

$2 (6 x^{2}) + 2 (x) + 4 = 0$

Factor $2$ out of $4$ .

$2 (6 x^{2}) + 2 x + 2 \cdot 2 = 0$

Factor $2$ out of $2 (6 x^{2}) + 2 x$ .

$2 (6 x^{2} + x) + 2 \cdot 2 = 0$

Factor $2$ out of $2 (6 x^{2} + x) + 2 \cdot 2$ .

$2 (6 x^{2} + x + 2) = 0$

Divide each term in $2 (6 x^{2} + x + 2) = 0$ by $2$ and simplify.

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Divide each term in $2 (6 x^{2} + x + 2) = 0$ by $2$ .

$\frac{2 (6 x^{2} + x + 2)}{2} = \frac{0}{2}$

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 (6 x^{2} + x + 2)}{2} = \frac{0}{2}$

Divide $6 x^{2} + x + 2$ by $1$ .

$6 x^{2} + x + 2 = \frac{0}{2}$

Simplify the right side.

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Divide $0$ by $2$ .

$6 x^{2} + x + 2 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 6$ , $b = 1$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (6 \cdot 2)}}{2 \cdot 6}$

Simplify.

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Simplify the numerator.

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One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 6 \cdot 2}}{2 \cdot 6}$

Multiply $- 4 \cdot 6 \cdot 2$ .

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Multiply $- 4$ by $6$ .

$x = \frac{- 1 \pm \sqrt{1 - 24 \cdot 2}}{2 \cdot 6}$

Multiply $- 24$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 48}}{2 \cdot 6}$

Subtract $48$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 47}}{2 \cdot 6}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 6}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 6}$

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

$x = \frac{- 1 \pm i \sqrt{47}}{2 \cdot 6}$

Multiply $2$ by $6$ .

$x = \frac{- 1 \pm i \sqrt{47}}{12}$

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

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Simplify the numerator.

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One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 6 \cdot 2}}{2 \cdot 6}$

Multiply $- 4 \cdot 6 \cdot 2$ .

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Multiply $- 4$ by $6$ .

$x = \frac{- 1 \pm \sqrt{1 - 24 \cdot 2}}{2 \cdot 6}$

Multiply $- 24$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 48}}{2 \cdot 6}$

Subtract $48$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 47}}{2 \cdot 6}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 6}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 6}$

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

$x = \frac{- 1 \pm i \sqrt{47}}{2 \cdot 6}$

Multiply $2$ by $6$ .

$x = \frac{- 1 \pm i \sqrt{47}}{12}$

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{47}}{12}$

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

$x = \frac{- 1 \cdot 1 + i \sqrt{47}}{12}$

Factor $- 1$ out of $i \sqrt{47}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{47})}{12}$

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{47})$ .

$x = \frac{- 1 (1 - i \sqrt{47})}{12}$

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{47}}{12}$

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

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Simplify the numerator.

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One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 6 \cdot 2}}{2 \cdot 6}$

Multiply $- 4 \cdot 6 \cdot 2$ .

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Multiply $- 4$ by $6$ .

$x = \frac{- 1 \pm \sqrt{1 - 24 \cdot 2}}{2 \cdot 6}$

Multiply $- 24$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 48}}{2 \cdot 6}$

Subtract $48$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 47}}{2 \cdot 6}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 6}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 6}$

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

$x = \frac{- 1 \pm i \sqrt{47}}{2 \cdot 6}$

Multiply $2$ by $6$ .

$x = \frac{- 1 \pm i \sqrt{47}}{12}$

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{47}}{12}$

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

$x = \frac{- 1 \cdot 1 - i \sqrt{47}}{12}$

Factor $- 1$ out of $- i \sqrt{47}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{47})}{12}$

Factor $- 1$ out of $- 1 (1) - (i \sqrt{47})$ .

$x = \frac{- 1 (1 + i \sqrt{47})}{12}$

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{47}}{12}$

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{47}}{12}, - \frac{1 + i \sqrt{47}}{12}$

Step 3

Find the values where the derivative is undefined.

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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