Calculus Examples

$16 x^{4} + 125 x$

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 $16 x^{4} + 125 x$ with respect to $x$ is $\frac{d}{d x} [16 x^{4}] + \frac{d}{d x} [125 x]$ .

$f' (x) = \frac{d}{d x} (16 x^{4}) + \frac{d}{d x} (125 x)$

Step 1.1.2

Evaluate $\frac{d}{d x} [16 x^{4}]$ .

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Step 1.1.2.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{4}$ with respect to $x$ is $16 \frac{d}{d x} [x^{4}]$ .

$f' (x) = 16 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (125 x)$

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 $n = 4$ .

$f' (x) = 16 (4 x^{3}) + \frac{d}{d x} (125 x)$

Step 1.1.2.3

Multiply $4$ by $16$ .

$f' (x) = 64 x^{3} + \frac{d}{d x} (125 x)$

Step 1.1.3

Evaluate $\frac{d}{d x} [125 x]$ .

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Step 1.1.3.1

Since $125$ is constant with respect to $x$ , the derivative of $125 x$ with respect to $x$ is $125 \frac{d}{d x} [x]$ .

$f' (x) = 64 x^{3} + 125 \frac{d}{d x} (x)$

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 $n = 1$ .

$f' (x) = 64 x^{3} + 125 \cdot 1$

Step 1.1.3.3

Multiply $125$ by $1$ .

$f' (x) = 64 x^{3} + 125$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $64 x^{3} + 125$ .

$64 x^{3} + 125$

Step 2

Set the first derivative equal to $0$ then solve the equation $64 x^{3} + 125 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$64 x^{3} + 125 = 0$

Step 2.2

Subtract $125$ from both sides of the equation.

$64 x^{3} = - 125$

Step 2.3

Add $125$ to both sides of the equation.

$64 x^{3} + 125 = 0$

Step 2.4

Factor the left side of the equation.

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Step 2.4.1

Rewrite $64 x^{3}$ as ${(4 x)}^{3}$ .

${(4 x)}^{3} + 125 = 0$

Step 2.4.2

Rewrite $125$ as $5^{3}$ .

${(4 x)}^{3} + 5^{3} = 0$

Step 2.4.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 4 x$ and $b = 5$ .

$(4 x + 5) ({(4 x)}^{2} - 4 x \cdot 5 + 5^{2}) = 0$

Step 2.4.4

Simplify.

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Step 2.4.4.1

Apply the product rule to $4 x$ .

$(4 x + 5) (4^{2} x^{2} - 4 x \cdot 5 + 5^{2}) = 0$

Step 2.4.4.2

Raise $4$ to the power of $2$ .

$(4 x + 5) (16 x^{2} - 4 x \cdot 5 + 5^{2}) = 0$

Step 2.4.4.3

Multiply $4$ by $- 1$ .

$(4 x + 5) (16 x^{2} - 4 x \cdot 5 + 5^{2}) = 0$

Step 2.4.4.4

Multiply $5$ by $- 4$ .

$(4 x + 5) (16 x^{2} - 20 x + 5^{2}) = 0$

Step 2.4.4.5

Raise $5$ to the power of $2$ .

$(4 x + 5) (16 x^{2} - 20 x + 25) = 0$

Step 2.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$4 x + 5 = 0$

$16 x^{2} - 20 x + 25 = 0$

Step 2.6

Set $4 x + 5$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $4 x + 5$ equal to $0$ .

$4 x + 5 = 0$

Step 2.6.2

Solve $4 x + 5 = 0$ for $x$ .

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Step 2.6.2.1

Subtract $5$ from both sides of the equation.

$4 x = - 5$

Step 2.6.2.2

Divide each term in $4 x = - 5$ by $4$ and simplify.

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Step 2.6.2.2.1

Divide each term in $4 x = - 5$ by $4$ .

$\frac{4 x}{4} = \frac{- 5}{4}$

Step 2.6.2.2.2

Simplify the left side.

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Step 2.6.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.6.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 5}{4}$

Step 2.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{4}$

Step 2.6.2.2.3

Simplify the right side.

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Step 2.6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{4}$

Step 2.7

Set $16 x^{2} - 20 x + 25$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $16 x^{2} - 20 x + 25$ equal to $0$ .

$16 x^{2} - 20 x + 25 = 0$

Step 2.7.2

Solve $16 x^{2} - 20 x + 25 = 0$ for $x$ .

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Step 2.7.2.1

Use the quadratic formula to find the solutions.

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

Step 2.7.2.2

Substitute the values $a = 16$ , $b = - 20$ , and $c = 25$ into the quadratic formula and solve for $x$ .

$\frac{20 \pm \sqrt{{(- 20)}^{2} - 4 \cdot (16 \cdot 25)}}{2 \cdot 16}$

Step 2.7.2.3

Simplify.

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Step 2.7.2.3.1

Simplify the numerator.

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Step 2.7.2.3.1.1

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 16 \cdot 25}}{2 \cdot 16}$

Step 2.7.2.3.1.2

Multiply $- 4 \cdot 16 \cdot 25$ .

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Step 2.7.2.3.1.2.1

Multiply $- 4$ by $16$ .

$x = \frac{20 \pm \sqrt{400 - 64 \cdot 25}}{2 \cdot 16}$

Step 2.7.2.3.1.2.2

Multiply $- 64$ by $25$ .

$x = \frac{20 \pm \sqrt{400 - 1600}}{2 \cdot 16}$

Step 2.7.2.3.1.3

Subtract $1600$ from $400$ .

$x = \frac{20 \pm \sqrt{- 1200}}{2 \cdot 16}$

Step 2.7.2.3.1.4

Rewrite $- 1200$ as $- 1 (1200)$ .

$x = \frac{20 \pm \sqrt{- 1 \cdot 1200}}{2 \cdot 16}$

Step 2.7.2.3.1.5

Rewrite $\sqrt{- 1 (1200)}$ as $\sqrt{- 1} \cdot \sqrt{1200}$ .

$x = \frac{20 \pm \sqrt{- 1} \cdot \sqrt{1200}}{2 \cdot 16}$

Step 2.7.2.3.1.6

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

$x = \frac{20 \pm i \cdot \sqrt{1200}}{2 \cdot 16}$

Step 2.7.2.3.1.7

Rewrite $1200$ as $20^{2} \cdot 3$ .

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Step 2.7.2.3.1.7.1

Factor $400$ out of $1200$ .

$x = \frac{20 \pm i \cdot \sqrt{400 (3)}}{2 \cdot 16}$

Step 2.7.2.3.1.7.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{20 \pm i \cdot \sqrt{20^{2} \cdot 3}}{2 \cdot 16}$

Step 2.7.2.3.1.8

Pull terms out from under the radical.

$x = \frac{20 \pm i \cdot (20 \sqrt{3})}{2 \cdot 16}$

Step 2.7.2.3.1.9

Move $20$ to the left of $i$ .

$x = \frac{20 \pm 20 i \sqrt{3}}{2 \cdot 16}$

Step 2.7.2.3.2

Multiply $2$ by $16$ .

$x = \frac{20 \pm 20 i \sqrt{3}}{32}$

Step 2.7.2.3.3

Simplify $\frac{20 \pm 20 i \sqrt{3}}{32}$ .

$x = \frac{5 \pm 5 i \sqrt{3}}{8}$

Step 2.7.2.4

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

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Step 2.7.2.4.1

Simplify the numerator.

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Step 2.7.2.4.1.1

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 16 \cdot 25}}{2 \cdot 16}$

Step 2.7.2.4.1.2

Multiply $- 4 \cdot 16 \cdot 25$ .

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Step 2.7.2.4.1.2.1

Multiply $- 4$ by $16$ .

$x = \frac{20 \pm \sqrt{400 - 64 \cdot 25}}{2 \cdot 16}$

Step 2.7.2.4.1.2.2

Multiply $- 64$ by $25$ .

$x = \frac{20 \pm \sqrt{400 - 1600}}{2 \cdot 16}$

Step 2.7.2.4.1.3

Subtract $1600$ from $400$ .

$x = \frac{20 \pm \sqrt{- 1200}}{2 \cdot 16}$

Step 2.7.2.4.1.4

Rewrite $- 1200$ as $- 1 (1200)$ .

$x = \frac{20 \pm \sqrt{- 1 \cdot 1200}}{2 \cdot 16}$

Step 2.7.2.4.1.5

Rewrite $\sqrt{- 1 (1200)}$ as $\sqrt{- 1} \cdot \sqrt{1200}$ .

$x = \frac{20 \pm \sqrt{- 1} \cdot \sqrt{1200}}{2 \cdot 16}$

Step 2.7.2.4.1.6

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

$x = \frac{20 \pm i \cdot \sqrt{1200}}{2 \cdot 16}$

Step 2.7.2.4.1.7

Rewrite $1200$ as $20^{2} \cdot 3$ .

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Step 2.7.2.4.1.7.1

Factor $400$ out of $1200$ .

$x = \frac{20 \pm i \cdot \sqrt{400 (3)}}{2 \cdot 16}$

Step 2.7.2.4.1.7.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{20 \pm i \cdot \sqrt{20^{2} \cdot 3}}{2 \cdot 16}$

Step 2.7.2.4.1.8

Pull terms out from under the radical.

$x = \frac{20 \pm i \cdot (20 \sqrt{3})}{2 \cdot 16}$

Step 2.7.2.4.1.9

Move $20$ to the left of $i$ .

$x = \frac{20 \pm 20 i \sqrt{3}}{2 \cdot 16}$

Step 2.7.2.4.2

Multiply $2$ by $16$ .

$x = \frac{20 \pm 20 i \sqrt{3}}{32}$

Step 2.7.2.4.3

Simplify $\frac{20 \pm 20 i \sqrt{3}}{32}$ .

$x = \frac{5 \pm 5 i \sqrt{3}}{8}$

Step 2.7.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{5 + 5 i \sqrt{3}}{8}$

Step 2.7.2.5

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

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Step 2.7.2.5.1

Simplify the numerator.

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Step 2.7.2.5.1.1

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 16 \cdot 25}}{2 \cdot 16}$

Step 2.7.2.5.1.2

Multiply $- 4 \cdot 16 \cdot 25$ .

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Step 2.7.2.5.1.2.1

Multiply $- 4$ by $16$ .

$x = \frac{20 \pm \sqrt{400 - 64 \cdot 25}}{2 \cdot 16}$

Step 2.7.2.5.1.2.2

Multiply $- 64$ by $25$ .

$x = \frac{20 \pm \sqrt{400 - 1600}}{2 \cdot 16}$

Step 2.7.2.5.1.3

Subtract $1600$ from $400$ .

$x = \frac{20 \pm \sqrt{- 1200}}{2 \cdot 16}$

Step 2.7.2.5.1.4

Rewrite $- 1200$ as $- 1 (1200)$ .

$x = \frac{20 \pm \sqrt{- 1 \cdot 1200}}{2 \cdot 16}$

Step 2.7.2.5.1.5

Rewrite $\sqrt{- 1 (1200)}$ as $\sqrt{- 1} \cdot \sqrt{1200}$ .

$x = \frac{20 \pm \sqrt{- 1} \cdot \sqrt{1200}}{2 \cdot 16}$

Step 2.7.2.5.1.6

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

$x = \frac{20 \pm i \cdot \sqrt{1200}}{2 \cdot 16}$

Step 2.7.2.5.1.7

Rewrite $1200$ as $20^{2} \cdot 3$ .

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Step 2.7.2.5.1.7.1

Factor $400$ out of $1200$ .

$x = \frac{20 \pm i \cdot \sqrt{400 (3)}}{2 \cdot 16}$

Step 2.7.2.5.1.7.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{20 \pm i \cdot \sqrt{20^{2} \cdot 3}}{2 \cdot 16}$

Step 2.7.2.5.1.8

Pull terms out from under the radical.

$x = \frac{20 \pm i \cdot (20 \sqrt{3})}{2 \cdot 16}$

Step 2.7.2.5.1.9

Move $20$ to the left of $i$ .

$x = \frac{20 \pm 20 i \sqrt{3}}{2 \cdot 16}$

Step 2.7.2.5.2

Multiply $2$ by $16$ .

$x = \frac{20 \pm 20 i \sqrt{3}}{32}$

Step 2.7.2.5.3

Simplify $\frac{20 \pm 20 i \sqrt{3}}{32}$ .

$x = \frac{5 \pm 5 i \sqrt{3}}{8}$

Step 2.7.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{5 - 5 i \sqrt{3}}{8}$

Step 2.7.2.6

The final answer is the combination of both solutions.

$x = \frac{5 + 5 i \sqrt{3}}{8}, \frac{5 - 5 i \sqrt{3}}{8}$

Step 2.8

The final solution is all the values that make $(4 x + 5) (16 x^{2} - 20 x + 25) = 0$ true.

$x = - \frac{5}{4}, \frac{5 + 5 i \sqrt{3}}{8}, \frac{5 - 5 i \sqrt{3}}{8}$

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 $16 x^{4} + 125 x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - \frac{5}{4}$ .

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Step 4.1.1

Substitute $- \frac{5}{4}$ for $x$ .

$16 {(- \frac{5}{4})}^{4} + 125 (- \frac{5}{4})$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.1.1.1

Apply the product rule to $- \frac{5}{4}$ .

$16 ({(- 1)}^{4} {(\frac{5}{4})}^{4}) + 125 (- \frac{5}{4})$

Step 4.1.2.1.1.2

Apply the product rule to $\frac{5}{4}$ .

$16 ({(- 1)}^{4} \frac{5^{4}}{4^{4}}) + 125 (- \frac{5}{4})$

Step 4.1.2.1.2

Raise $- 1$ to the power of $4$ .

$16 (1 \frac{5^{4}}{4^{4}}) + 125 (- \frac{5}{4})$

Step 4.1.2.1.3

Multiply $\frac{5^{4}}{4^{4}}$ by $1$ .

$16 \frac{5^{4}}{4^{4}} + 125 (- \frac{5}{4})$

Step 4.1.2.1.4

Raise $5$ to the power of $4$ .

$16 \frac{625}{4^{4}} + 125 (- \frac{5}{4})$

Step 4.1.2.1.5

Raise $4$ to the power of $4$ .

$16 (\frac{625}{256}) + 125 (- \frac{5}{4})$

Step 4.1.2.1.6

Cancel the common factor of $16$ .

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Step 4.1.2.1.6.1

Factor $16$ out of $256$ .

$16 \frac{625}{16 (16)} + 125 (- \frac{5}{4})$

Step 4.1.2.1.6.2

Cancel the common factor.

$16 \frac{625}{16 \cdot 16} + 125 (- \frac{5}{4})$

Step 4.1.2.1.6.3

Rewrite the expression.

$\frac{625}{16} + 125 (- \frac{5}{4})$

Step 4.1.2.1.7

Multiply $125 (- \frac{5}{4})$ .

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Step 4.1.2.1.7.1

Multiply $- 1$ by $125$ .

$\frac{625}{16} - 125 (\frac{5}{4})$

Step 4.1.2.1.7.2

Combine $- 125$ and $\frac{5}{4}$ .

$\frac{625}{16} + \frac{- 125 \cdot 5}{4}$

Step 4.1.2.1.7.3

Multiply $- 125$ by $5$ .

$\frac{625}{16} + \frac{- 625}{4}$

Step 4.1.2.1.8

Move the negative in front of the fraction.

$\frac{625}{16} - \frac{625}{4}$

Step 4.1.2.2

To write $- \frac{625}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{625}{16} - \frac{625}{4} \cdot \frac{4}{4}$

Step 4.1.2.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.3.1

Multiply $\frac{625}{4}$ by $\frac{4}{4}$ .

$\frac{625}{16} - \frac{625 \cdot 4}{4 \cdot 4}$

Step 4.1.2.3.2

Multiply $4$ by $4$ .

$\frac{625}{16} - \frac{625 \cdot 4}{16}$

Step 4.1.2.4

Combine the numerators over the common denominator.

$\frac{625 - 625 \cdot 4}{16}$

Step 4.1.2.5

Simplify the numerator.

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Step 4.1.2.5.1

Multiply $- 625$ by $4$ .

$\frac{625 - 2500}{16}$

Step 4.1.2.5.2

Subtract $2500$ from $625$ .

$\frac{- 1875}{16}$

Step 4.1.2.6

Move the negative in front of the fraction.

$- \frac{1875}{16}$

Step 4.2

List all of the points.

$(- \frac{5}{4}, - \frac{1875}{16})$

Step 5