Calculus Examples

$f (x) = 5 \tan^{-1} (x) - 3 x^{3}$

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

By the Sum Rule, the derivative of $5 \tan^{-1} (x) - 3 x^{3}$ with respect to $x$ is $\frac{d}{d x} [5 \tan^{-1} (x)] + \frac{d}{d x} [- 3 x^{3}]$ .

$f' (x) = \frac{d}{d x} (5 \tan^{-1} (x)) + \frac{d}{d x} (- 3 x^{3})$

Step 1.1.2

Evaluate $\frac{d}{d x} [5 \tan^{-1} (x)]$ .

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Step 1.1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \tan^{-1} (x)$ with respect to $x$ is $5 \frac{d}{d x} [\tan^{-1} (x)]$ .

$f' (x) = 5 \frac{d}{d x} (\tan^{-1} (x)) + \frac{d}{d x} (- 3 x^{3})$

Step 1.1.2.2

The derivative of $\tan^{-1} (x)$ with respect to $x$ is $\frac{1}{1 + x^{2}}$ .

$f' (x) = 5 (\frac{1}{1 + x^{2}}) + \frac{d}{d x} (- 3 x^{3})$

Step 1.1.2.3

Combine $5$ and $\frac{1}{1 + x^{2}}$ .

$f' (x) = \frac{5}{1 + x^{2}} + \frac{d}{d x} (- 3 x^{3})$

Step 1.1.3

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

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Step 1.1.3.1

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

$f' (x) = \frac{5}{1 + x^{2}} - 3 \frac{d}{d x} x^{3}$

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 $n = 3$ .

$f' (x) = \frac{5}{1 + x^{2}} - 3 (3 x^{2})$

Step 1.1.3.3

Multiply $3$ by $- 3$ .

$f' (x) = \frac{5}{1 + x^{2}} - 9 x^{2}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Combine terms.

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Step 1.1.4.1.1

To write $- 9 x^{2}$ as a fraction with a common denominator, multiply by $\frac{1 + x^{2}}{1 + x^{2}}$ .

$f' (x) = \frac{5}{1 + x^{2}} - 9 x^{2} \cdot \frac{1 + x^{2}}{1 + x^{2}}$

Step 1.1.4.1.2

Combine $- 9 x^{2}$ and $\frac{1 + x^{2}}{1 + x^{2}}$ .

$f' (x) = \frac{5}{1 + x^{2}} + \frac{- 9 x^{2} (1 + x^{2})}{1 + x^{2}}$

Step 1.1.4.1.3

Combine the numerators over the common denominator.

$f' (x) = \frac{5 - 9 x^{2} (1 + x^{2})}{1 + x^{2}}$

Step 1.1.4.2

Reorder terms.

$f' (x) = \frac{- 9 x^{2} (1 + x^{2}) + 5}{x^{2} + 1}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{- 9 x^{2} (1 + x^{2}) + 5}{x^{2} + 1}$ .

$\frac{- 9 x^{2} (1 + x^{2}) + 5}{x^{2} + 1}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{- 9 x^{2} (1 + x^{2}) + 5}{x^{2} + 1} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{- 9 x^{2} (1 + x^{2}) + 5}{x^{2} + 1} = 0$

Step 2.2

Set the numerator equal to zero.

$- 9 x^{2} (1 + x^{2}) + 5 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Apply the distributive property.

$- 9 x^{2} \cdot 1 - 9 x^{2} x^{2} + 5 = 0$

Step 2.3.1.2

Multiply $- 9$ by $1$ .

$- 9 x^{2} - 9 x^{2} x^{2} + 5 = 0$

Step 2.3.1.3

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.3.1.3.1

Move $x^{2}$ .

$- 9 x^{2} - 9 (x^{2} x^{2}) + 5 = 0$

Step 2.3.1.3.2

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

$- 9 x^{2} - 9 x^{2 + 2} + 5 = 0$

Step 2.3.1.3.3

Add $2$ and $2$ .

$- 9 x^{2} - 9 x^{4} + 5 = 0$

Step 2.3.2

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$- 9 u^{2} - 9 u + 5 = 0$

$u = x^{2}$

Step 2.3.3

Use the quadratic formula to find the solutions.

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

Step 2.3.4

Substitute the values $a = - 9$ , $b = - 9$ , and $c = 5$ into the quadratic formula and solve for $u$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (- 9 \cdot 5)}}{2 \cdot - 9}$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Simplify the numerator.

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Step 2.3.5.1.1

Raise $- 9$ to the power of $2$ .

$u = \frac{9 \pm \sqrt{81 - 4 \cdot - 9 \cdot 5}}{2 \cdot - 9}$

Step 2.3.5.1.2

Multiply $- 4 \cdot - 9 \cdot 5$ .

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Step 2.3.5.1.2.1

Multiply $- 4$ by $- 9$ .

$u = \frac{9 \pm \sqrt{81 + 36 \cdot 5}}{2 \cdot - 9}$

Step 2.3.5.1.2.2

Multiply $36$ by $5$ .

$u = \frac{9 \pm \sqrt{81 + 180}}{2 \cdot - 9}$

Step 2.3.5.1.3

Add $81$ and $180$ .

$u = \frac{9 \pm \sqrt{261}}{2 \cdot - 9}$

Step 2.3.5.1.4

Rewrite $261$ as $3^{2} \cdot 29$ .

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Step 2.3.5.1.4.1

Factor $9$ out of $261$ .

$u = \frac{9 \pm \sqrt{9 (29)}}{2 \cdot - 9}$

Step 2.3.5.1.4.2

Rewrite $9$ as $3^{2}$ .

$u = \frac{9 \pm \sqrt{3^{2} \cdot 29}}{2 \cdot - 9}$

Step 2.3.5.1.5

Pull terms out from under the radical.

$u = \frac{9 \pm 3 \sqrt{29}}{2 \cdot - 9}$

Step 2.3.5.2

Multiply $2$ by $- 9$ .

$u = \frac{9 \pm 3 \sqrt{29}}{- 18}$

Step 2.3.5.3

Simplify $\frac{9 \pm 3 \sqrt{29}}{- 18}$ .

$u = \frac{3 \pm \sqrt{29}}{- 6}$

Step 2.3.5.4

Move the negative in front of the fraction.

$u = - \frac{3 \pm \sqrt{29}}{6}$

Step 2.3.6

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

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Step 2.3.6.1

Simplify the numerator.

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Step 2.3.6.1.1

Raise $- 9$ to the power of $2$ .

$u = \frac{9 \pm \sqrt{81 - 4 \cdot - 9 \cdot 5}}{2 \cdot - 9}$

Step 2.3.6.1.2

Multiply $- 4 \cdot - 9 \cdot 5$ .

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Step 2.3.6.1.2.1

Multiply $- 4$ by $- 9$ .

$u = \frac{9 \pm \sqrt{81 + 36 \cdot 5}}{2 \cdot - 9}$

Step 2.3.6.1.2.2

Multiply $36$ by $5$ .

$u = \frac{9 \pm \sqrt{81 + 180}}{2 \cdot - 9}$

Step 2.3.6.1.3

Add $81$ and $180$ .

$u = \frac{9 \pm \sqrt{261}}{2 \cdot - 9}$

Step 2.3.6.1.4

Rewrite $261$ as $3^{2} \cdot 29$ .

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Step 2.3.6.1.4.1

Factor $9$ out of $261$ .

$u = \frac{9 \pm \sqrt{9 (29)}}{2 \cdot - 9}$

Step 2.3.6.1.4.2

Rewrite $9$ as $3^{2}$ .

$u = \frac{9 \pm \sqrt{3^{2} \cdot 29}}{2 \cdot - 9}$

Step 2.3.6.1.5

Pull terms out from under the radical.

$u = \frac{9 \pm 3 \sqrt{29}}{2 \cdot - 9}$

Step 2.3.6.2

Multiply $2$ by $- 9$ .

$u = \frac{9 \pm 3 \sqrt{29}}{- 18}$

Step 2.3.6.3

Simplify $\frac{9 \pm 3 \sqrt{29}}{- 18}$ .

$u = \frac{3 \pm \sqrt{29}}{- 6}$

Step 2.3.6.4

Move the negative in front of the fraction.

$u = - \frac{3 \pm \sqrt{29}}{6}$

Step 2.3.6.5

Change the $\pm$ to $+$ .

$u = - \frac{3 + \sqrt{29}}{6}$

Step 2.3.7

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

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Step 2.3.7.1

Simplify the numerator.

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Step 2.3.7.1.1

Raise $- 9$ to the power of $2$ .

$u = \frac{9 \pm \sqrt{81 - 4 \cdot - 9 \cdot 5}}{2 \cdot - 9}$

Step 2.3.7.1.2

Multiply $- 4 \cdot - 9 \cdot 5$ .

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Step 2.3.7.1.2.1

Multiply $- 4$ by $- 9$ .

$u = \frac{9 \pm \sqrt{81 + 36 \cdot 5}}{2 \cdot - 9}$

Step 2.3.7.1.2.2

Multiply $36$ by $5$ .

$u = \frac{9 \pm \sqrt{81 + 180}}{2 \cdot - 9}$

Step 2.3.7.1.3

Add $81$ and $180$ .

$u = \frac{9 \pm \sqrt{261}}{2 \cdot - 9}$

Step 2.3.7.1.4

Rewrite $261$ as $3^{2} \cdot 29$ .

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Step 2.3.7.1.4.1

Factor $9$ out of $261$ .

$u = \frac{9 \pm \sqrt{9 (29)}}{2 \cdot - 9}$

Step 2.3.7.1.4.2

Rewrite $9$ as $3^{2}$ .

$u = \frac{9 \pm \sqrt{3^{2} \cdot 29}}{2 \cdot - 9}$

Step 2.3.7.1.5

Pull terms out from under the radical.

$u = \frac{9 \pm 3 \sqrt{29}}{2 \cdot - 9}$

Step 2.3.7.2

Multiply $2$ by $- 9$ .

$u = \frac{9 \pm 3 \sqrt{29}}{- 18}$

Step 2.3.7.3

Simplify $\frac{9 \pm 3 \sqrt{29}}{- 18}$ .

$u = \frac{3 \pm \sqrt{29}}{- 6}$

Step 2.3.7.4

Move the negative in front of the fraction.

$u = - \frac{3 \pm \sqrt{29}}{6}$

Step 2.3.7.5

Change the $\pm$ to $-$ .

$u = - \frac{3 - \sqrt{29}}{6}$

Step 2.3.8

The final answer is the combination of both solutions.

$u = - \frac{3 + \sqrt{29}}{6}, - \frac{3 - \sqrt{29}}{6}$

Step 2.3.9

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = - 1.39752746$

${(x^{2})}^{1} = 0.39752746$

Step 2.3.10

Solve the first equation for $x$ .

$x^{2} = - 1.39752746$

Step 2.3.11

Solve the equation for $x$ .

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Step 2.3.11.1

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

$x = \pm \sqrt{- 1.39752746}$

Step 2.3.11.2

Simplify $\pm \sqrt{- 1.39752746}$ .

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Step 2.3.11.2.1

Rewrite $- 1.39752746$ as $- 1 (1.39752746)$ .

$x = \pm \sqrt{- 1 (1.39752746)}$

Step 2.3.11.2.2

Rewrite $\sqrt{- 1 (1.39752746)}$ as $\sqrt{- 1} \cdot \sqrt{1.39752746}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{1.39752746}$

Step 2.3.11.2.3

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

$x = \pm i \sqrt{1.39752746}$

Step 2.3.11.3

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

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Step 2.3.11.3.1

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

$x = i \sqrt{1.39752746}$

Step 2.3.11.3.2

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

$x = - i \sqrt{1.39752746}$

Step 2.3.11.3.3

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

$x = i \sqrt{1.39752746}, - i \sqrt{1.39752746}$

Step 2.3.12

Solve the second equation for $x$ .

${(x^{2})}^{1} = 0.39752746$

Step 2.3.13

Solve the equation for $x$ .

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Step 2.3.13.1

Remove parentheses.

$x^{2} = 0.39752746$

Step 2.3.13.2

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

$x = \pm \sqrt{0.39752746}$

Step 2.3.13.3

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

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Step 2.3.13.3.1

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

$x = \sqrt{0.39752746}$

Step 2.3.13.3.2

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

$x = - \sqrt{0.39752746}$

Step 2.3.13.3.3

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

$x = \sqrt{0.39752746}, - \sqrt{0.39752746}$

Step 2.3.14

The solution to $- 9 x^{4} - 9 x^{2} + 5 = 0$ is $x = i \sqrt{1.39752746}, - i \sqrt{1.39752746}, \sqrt{0.39752746}, - \sqrt{0.39752746}$ .

$x = i \sqrt{1.39752746}, - i \sqrt{1.39752746}, \sqrt{0.39752746}, - \sqrt{0.39752746}$

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

Evaluate $5 \tan^{-1} (x) - 3 x^{3}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \sqrt{0.39752746}$ .

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Step 4.1.1

Substitute $\sqrt{0.39752746}$ for $x$ .

$5 \tan^{-1} (\sqrt{0.39752746}) - 3 {(\sqrt{0.39752746})}^{3}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Evaluate $\tan^{-1} (\sqrt{0.39752746})$ .

$5 \cdot 0.56254301 - 3 {(\sqrt{0.39752746})}^{3}$

Step 4.1.2.1.2

Multiply $5$ by $0.56254301$ .

$2.81271509 - 3 {(\sqrt{0.39752746})}^{3}$

Step 4.1.2.1.3

Rewrite ${\sqrt{0.39752746}}^{3}$ as $\sqrt{{0.39752746}^{3}}$ .

$2.81271509 - 3 \sqrt{{0.39752746}^{3}}$

Step 4.1.2.1.4

Raise $0.39752746$ to the power of $3$ .

$2.81271509 - 3 \sqrt{0.0628205}$

Step 4.1.2.2

Subtract $3 \sqrt{0.0628205}$ from $2.81271509$ .

$2.06079452$

Step 4.2

Evaluate at $x = - \sqrt{0.39752746}$ .

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Step 4.2.1

Substitute $- \sqrt{0.39752746}$ for $x$ .

$5 \tan^{-1} (- \sqrt{0.39752746}) - 3 {(- \sqrt{0.39752746})}^{3}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Evaluate $\tan^{-1} (- \sqrt{0.39752746})$ .

$5 \cdot - 0.56254301 - 3 {(- \sqrt{0.39752746})}^{3}$

Step 4.2.2.1.2

Multiply $5$ by $- 0.56254301$ .

$- 2.81271509 - 3 {(- \sqrt{0.39752746})}^{3}$

Step 4.2.2.1.3

Apply the product rule to $- \sqrt{0.39752746}$ .

$- 2.81271509 - 3 ({(- 1)}^{3} {\sqrt{0.39752746}}^{3})$

Step 4.2.2.1.4

Raise $- 1$ to the power of $3$ .

$- 2.81271509 - 3 (- {\sqrt{0.39752746}}^{3})$

Step 4.2.2.1.5

Rewrite ${\sqrt{0.39752746}}^{3}$ as $\sqrt{{0.39752746}^{3}}$ .

$- 2.81271509 - 3 (- \sqrt{{0.39752746}^{3}})$

Step 4.2.2.1.6

Raise $0.39752746$ to the power of $3$ .

$- 2.81271509 - 3 (- \sqrt{0.0628205})$

Step 4.2.2.1.7

Multiply $- 1$ by $- 3$ .

$- 2.81271509 + 3 \sqrt{0.0628205}$

Step 4.2.2.2

Add $- 2.81271509$ and $3 \sqrt{0.0628205}$ .

$- 2.06079452$

Step 4.3

List all of the points.

$(\sqrt{0.39752746}, 2.06079452), (- \sqrt{0.39752746}, - 2.06079452)$

Step 5