Calculus Examples

$f (x) = \frac{1}{3} x^{3} + \frac{3}{2} x^{2} + 8 x - 4$

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 $\frac{1}{3} x^{3} + \frac{3}{2} x^{2} + 8 x - 4$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [\frac{3}{2} x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [- 4]$ .

$f' (x) = \frac{d}{d x} (\frac{1}{3} x^{3}) + \frac{d}{d x} (\frac{3}{2} x^{2}) + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{3} x^{3}]$ .

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Step 1.1.2.1

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

$f' (x) = \frac{1}{3} \cdot \frac{d}{d x} (x^{3}) + \frac{d}{d x} (\frac{3}{2} x^{2}) + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

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 $n = 3$ .

$f' (x) = \frac{1}{3} \cdot (3 x^{2}) + \frac{d}{d x} (\frac{3}{2} x^{2}) + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.2.3

Combine $3$ and $\frac{1}{3}$ .

$f' (x) = \frac{3}{3} x^{2} + \frac{d}{d x} (\frac{3}{2} x^{2}) + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.2.4

Combine $\frac{3}{3}$ and $x^{2}$ .

$f' (x) = \frac{3 x^{2}}{3} + \frac{d}{d x} (\frac{3}{2} x^{2}) + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.2.5

Cancel the common factor of $3$ .

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Step 1.1.2.5.1

Cancel the common factor.

$f' (x) = \frac{3 x^{2}}{3} + \frac{d}{d x} (\frac{3}{2} x^{2}) + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.2.5.2

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

$f' (x) = x^{2} + \frac{d}{d x} (\frac{3}{2} x^{2}) + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.3

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

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Step 1.1.3.1

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3}{2} x^{2}$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [x^{2}]$ .

$f' (x) = x^{2} + \frac{3}{2} \cdot \frac{d}{d x} (x^{2}) + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

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 $n = 2$ .

$f' (x) = x^{2} + \frac{3}{2} \cdot (2 x) + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.3.3

Combine $2$ and $\frac{3}{2}$ .

$f' (x) = x^{2} + \frac{2 \cdot 3}{2} x + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.3.4

Multiply $2$ by $3$ .

$f' (x) = x^{2} + \frac{6}{2} x + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.3.5

Combine $\frac{6}{2}$ and $x$ .

$f' (x) = x^{2} + \frac{6 x}{2} + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.3.6

Cancel the common factor of $6$ and $2$ .

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Step 1.1.3.6.1

Factor $2$ out of $6 x$ .

$f' (x) = x^{2} + \frac{2 (3 x)}{2} + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.3.6.2

Cancel the common factors.

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Step 1.1.3.6.2.1

Factor $2$ out of $2$ .

$f' (x) = x^{2} + \frac{2 (3 x)}{2 (1)} + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.3.6.2.2

Cancel the common factor.

$f' (x) = x^{2} + \frac{2 (3 x)}{2 \cdot 1} + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.3.6.2.3

Rewrite the expression.

$f' (x) = x^{2} + \frac{3 x}{1} + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

Step 1.1.3.6.2.4

Divide $3 x$ by $1$ .

$f' (x) = x^{2} + 3 x + \frac{d}{d x} (8 x) + \frac{d}{d x} (- 4)$

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

$f' (x) = x^{2} + 3 x + 8 \frac{d}{d x} (x) + \frac{d}{d x} (- 4)$

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 $n = 1$ .

$f' (x) = x^{2} + 3 x + 8 \cdot 1 + \frac{d}{d x} (- 4)$

Step 1.1.4.3

Multiply $8$ by $1$ .

$f' (x) = x^{2} + 3 x + 8 + \frac{d}{d x} (- 4)$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$f' (x) = x^{2} + 3 x + 8 + 0$

Step 1.1.5.2

Add $x^{2} + 3 x + 8$ and $0$ .

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $x^{2} + 3 x + 8$ .

$x^{2} + 3 x + 8$

Step 2

Set the first derivative equal to $0$ then solve the equation $x^{2} + 3 x + 8 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$x^{2} + 3 x + 8 = 0$

Step 2.2

Use the quadratic formula to find the solutions.

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

Step 2.3

Substitute the values $a = 1$ , $b = 3$ , and $c = 8$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 8)}}{2 \cdot 1}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Step 2.4.1.2

Multiply $- 4 \cdot 1 \cdot 8$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 8}}{2 \cdot 1}$

Step 2.4.1.2.2

Multiply $- 4$ by $8$ .

$x = \frac{- 3 \pm \sqrt{9 - 32}}{2 \cdot 1}$

Step 2.4.1.3

Subtract $32$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 23}}{2 \cdot 1}$

Step 2.4.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 1}$

Step 2.4.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 1}$

Step 2.4.1.6

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

$x = \frac{- 3 \pm i \sqrt{23}}{2 \cdot 1}$

Step 2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{23}}{2}$

Step 2.5

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

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- 4 \cdot 1 \cdot 8$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 8}}{2 \cdot 1}$

Step 2.5.1.2.2

Multiply $- 4$ by $8$ .

$x = \frac{- 3 \pm \sqrt{9 - 32}}{2 \cdot 1}$

Step 2.5.1.3

Subtract $32$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 23}}{2 \cdot 1}$

Step 2.5.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 1}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 1}$

Step 2.5.1.6

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

$x = \frac{- 3 \pm i \sqrt{23}}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{23}}{2}$

Step 2.5.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + i \sqrt{23}}{2}$

Step 2.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + i \sqrt{23}}{2}$

Step 2.5.5

Factor $- 1$ out of $i \sqrt{23}$ .

$x = \frac{- 1 \cdot 3 - (- i \sqrt{23})}{2}$

Step 2.5.6

Factor $- 1$ out of $- 1 (3) - (- i \sqrt{23})$ .

$x = \frac{- 1 (3 - i \sqrt{23})}{2}$

Step 2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 - i \sqrt{23}}{2}$

Step 2.6

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

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Step 2.6.1.2

Multiply $- 4 \cdot 1 \cdot 8$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 8}}{2 \cdot 1}$

Step 2.6.1.2.2

Multiply $- 4$ by $8$ .

$x = \frac{- 3 \pm \sqrt{9 - 32}}{2 \cdot 1}$

Step 2.6.1.3

Subtract $32$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 23}}{2 \cdot 1}$

Step 2.6.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 1}$

Step 2.6.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 1}$

Step 2.6.1.6

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

$x = \frac{- 3 \pm i \sqrt{23}}{2 \cdot 1}$

Step 2.6.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{23}}{2}$

Step 2.6.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - i \sqrt{23}}{2}$

Step 2.6.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - i \sqrt{23}}{2}$

Step 2.6.5

Factor $- 1$ out of $- i \sqrt{23}$ .

$x = \frac{- 1 \cdot 3 - (i \sqrt{23})}{2}$

Step 2.6.6

Factor $- 1$ out of $- 1 (3) - (i \sqrt{23})$ .

$x = \frac{- 1 (3 + i \sqrt{23})}{2}$

Step 2.6.7

Move the negative in front of the fraction.

$x = - \frac{3 + i \sqrt{23}}{2}$

Step 2.7

The final answer is the combination of both solutions.

$x = - \frac{3 - i \sqrt{23}}{2}, - \frac{3 + i \sqrt{23}}{2}$

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