Calculus Examples

$y = x - x^{5}$

Step 1

Reorder $x$ and $- x^{5}$ .

$y = - x^{5} + x$

Step 2

Set $y$ as a function of $x$ .

$f (x) = - x^{5} + x$

Step 3

Find the derivative.

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Step 3.1

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

$\frac{d}{d x} [- x^{5}] + \frac{d}{d x} [x]$

Step 3.2

Evaluate $\frac{d}{d x} [- x^{5}]$ .

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Step 3.2.1

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

$- \frac{d}{d x} [x^{5}] + \frac{d}{d x} [x]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$- (5 x^{4}) + \frac{d}{d x} [x]$

Step 3.2.3

Multiply $5$ by $- 1$ .

$- 5 x^{4} + \frac{d}{d x} [x]$

Step 3.3

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

$- 5 x^{4} + 1$

Step 4

Set the derivative equal to $0$ then solve the equation $- 5 x^{4} + 1 = 0$ .

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Step 4.1

Subtract $1$ from both sides of the equation.

$- 5 x^{4} = - 1$

Step 4.2

Divide each term in $- 5 x^{4} = - 1$ by $- 5$ and simplify.

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Step 4.2.1

Divide each term in $- 5 x^{4} = - 1$ by $- 5$ .

$\frac{- 5 x^{4}}{- 5} = \frac{- 1}{- 5}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 5$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 5 x^{4}}{- 5} = \frac{- 1}{- 5}$

Step 4.2.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{- 1}{- 5}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Dividing two negative values results in a positive value.

$x^{4} = \frac{1}{5}$

Step 4.3

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

$x = \pm \sqrt[4]{\frac{1}{5}}$

Step 4.4

Simplify $\pm \sqrt[4]{\frac{1}{5}}$ .

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Step 4.4.1

Rewrite $\sqrt[4]{\frac{1}{5}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{5}}$ .

$x = \pm \frac{\sqrt[4]{1}}{\sqrt[4]{5}}$

Step 4.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt[4]{5}}$

Step 4.4.3

Multiply $\frac{1}{\sqrt[4]{5}}$ by $\frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{3}}$ .

$x = \pm \frac{1}{\sqrt[4]{5}} \cdot \frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{3}}$

Step 4.4.4

Combine and simplify the denominator.

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Step 4.4.4.1

Multiply $\frac{1}{\sqrt[4]{5}}$ by $\frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{3}}$ .

$x = \pm \frac{{\sqrt[4]{5}}^{3}}{\sqrt[4]{5} {\sqrt[4]{5}}^{3}}$

Step 4.4.4.2

Raise $\sqrt[4]{5}$ to the power of $1$ .

$x = \pm \frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{1} {\sqrt[4]{5}}^{3}}$

Step 4.4.4.3

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

$x = \pm \frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{1 + 3}}$

Step 4.4.4.4

Add $1$ and $3$ .

$x = \pm \frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{4}}$

Step 4.4.4.5

Rewrite ${\sqrt[4]{5}}^{4}$ as $5$ .

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Step 4.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{5}$ as $5^{\frac{1}{4}}$ .

$x = \pm \frac{{\sqrt[4]{5}}^{3}}{{(5^{\frac{1}{4}})}^{4}}$

Step 4.4.4.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{{\sqrt[4]{5}}^{3}}{5^{\frac{1}{4} \cdot 4}}$

Step 4.4.4.5.3

Combine $\frac{1}{4}$ and $4$ .

$x = \pm \frac{{\sqrt[4]{5}}^{3}}{5^{\frac{4}{4}}}$

Step 4.4.4.5.4

Cancel the common factor of $4$ .

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Step 4.4.4.5.4.1

Cancel the common factor.

$x = \pm \frac{{\sqrt[4]{5}}^{3}}{5^{\frac{4}{4}}}$

Step 4.4.4.5.4.2

Rewrite the expression.

$x = \pm \frac{{\sqrt[4]{5}}^{3}}{5^{1}}$

Step 4.4.4.5.5

Evaluate the exponent.

$x = \pm \frac{{\sqrt[4]{5}}^{3}}{5}$

Step 4.4.5

Simplify the numerator.

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Step 4.4.5.1

Rewrite ${\sqrt[4]{5}}^{3}$ as $\sqrt[4]{5^{3}}$ .

$x = \pm \frac{\sqrt[4]{5^{3}}}{5}$

Step 4.4.5.2

Raise $5$ to the power of $3$ .

$x = \pm \frac{\sqrt[4]{125}}{5}$

Step 4.5

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

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Step 4.5.1

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

$x = \frac{\sqrt[4]{125}}{5}$

Step 4.5.2

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

$x = - \frac{\sqrt[4]{125}}{5}$

Step 4.5.3

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

$x = \frac{\sqrt[4]{125}}{5}, - \frac{\sqrt[4]{125}}{5}$

Step 5

Solve the original function $f (x) = - x^{5} + x$ at $x = \frac{\sqrt[4]{125}}{5}$ .

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Step 5.1

Replace the variable $x$ with $\frac{\sqrt[4]{125}}{5}$ in the expression.

$f (\frac{\sqrt[4]{125}}{5}) = - {(\frac{\sqrt[4]{125}}{5})}^{5} + \frac{\sqrt[4]{125}}{5}$

Step 5.2

Simplify the result.

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Step 5.2.1

Remove parentheses.

$f (\frac{\sqrt[4]{125}}{5}) = - {(\frac{\sqrt[4]{125}}{5})}^{5} + \frac{\sqrt[4]{125}}{5}$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Apply the product rule to $\frac{\sqrt[4]{125}}{5}$ .

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{{\sqrt[4]{125}}^{5}}{5^{5}} + \frac{\sqrt[4]{125}}{5}$

Step 5.2.2.2

Simplify the numerator.

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Step 5.2.2.2.1

Rewrite ${\sqrt[4]{125}}^{5}$ as $\sqrt[4]{125^{5}}$ .

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{\sqrt[4]{125^{5}}}{5^{5}} + \frac{\sqrt[4]{125}}{5}$

Step 5.2.2.2.2

Raise $125$ to the power of $5$ .

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{\sqrt[4]{30517578125}}{5^{5}} + \frac{\sqrt[4]{125}}{5}$

Step 5.2.2.2.3

Rewrite $30517578125$ as $125^{4} \cdot 125$ .

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Step 5.2.2.2.3.1

Factor $244140625$ out of $30517578125$ .

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{\sqrt[4]{244140625 (125)}}{5^{5}} + \frac{\sqrt[4]{125}}{5}$

Step 5.2.2.2.3.2

Rewrite $244140625$ as $125^{4}$ .

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{\sqrt[4]{125^{4} \cdot 125}}{5^{5}} + \frac{\sqrt[4]{125}}{5}$

Step 5.2.2.2.4

Pull terms out from under the radical.

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{125 \sqrt[4]{125}}{5^{5}} + \frac{\sqrt[4]{125}}{5}$

Step 5.2.2.3

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

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{125 \sqrt[4]{125}}{3125} + \frac{\sqrt[4]{125}}{5}$

Step 5.2.2.4

Cancel the common factor of $125$ and $3125$ .

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Step 5.2.2.4.1

Factor $125$ out of $125 \sqrt[4]{125}$ .

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{125 (\sqrt[4]{125})}{3125} + \frac{\sqrt[4]{125}}{5}$

Step 5.2.2.4.2

Cancel the common factors.

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Step 5.2.2.4.2.1

Factor $125$ out of $3125$ .

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{125 \sqrt[4]{125}}{125 \cdot 25} + \frac{\sqrt[4]{125}}{5}$

Step 5.2.2.4.2.2

Cancel the common factor.

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{125 \sqrt[4]{125}}{125 \cdot 25} + \frac{\sqrt[4]{125}}{5}$

Step 5.2.2.4.2.3

Rewrite the expression.

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{\sqrt[4]{125}}{25} + \frac{\sqrt[4]{125}}{5}$

Step 5.2.3

To write $\frac{\sqrt[4]{125}}{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{\sqrt[4]{125}}{25} + \frac{\sqrt[4]{125}}{5} \cdot \frac{5}{5}$

Step 5.2.4

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.4.1

Multiply $\frac{\sqrt[4]{125}}{5}$ by $\frac{5}{5}$ .

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{\sqrt[4]{125}}{25} + \frac{\sqrt[4]{125} \cdot 5}{5 \cdot 5}$

Step 5.2.4.2

Multiply $5$ by $5$ .

$f (\frac{\sqrt[4]{125}}{5}) = - \frac{\sqrt[4]{125}}{25} + \frac{\sqrt[4]{125} \cdot 5}{25}$

Step 5.2.5

Combine the numerators over the common denominator.

$f (\frac{\sqrt[4]{125}}{5}) = \frac{- \sqrt[4]{125} + \sqrt[4]{125} \cdot 5}{25}$

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

Move $5$ to the left of $\sqrt[4]{125}$ .

$f (\frac{\sqrt[4]{125}}{5}) = \frac{- \sqrt[4]{125} + 5 \cdot \sqrt[4]{125}}{25}$

Step 5.2.6.2

Add $- \sqrt[4]{125}$ and $5 \sqrt[4]{125}$ .

$f (\frac{\sqrt[4]{125}}{5}) = \frac{4 \sqrt[4]{125}}{25}$

Step 5.2.7

The final answer is $\frac{4 \sqrt[4]{125}}{25}$ .

$\frac{4 \sqrt[4]{125}}{25}$

Step 6

Solve the original function $f (x) = - x^{5} + x$ at $x = - \frac{\sqrt[4]{125}}{5}$ .

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Step 6.1

Replace the variable $x$ with $- \frac{\sqrt[4]{125}}{5}$ in the expression.

$f (- \frac{\sqrt[4]{125}}{5}) = - {(- \frac{\sqrt[4]{125}}{5})}^{5} - \frac{\sqrt[4]{125}}{5}$

Step 6.2

Simplify the result.

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Step 6.2.1

Remove parentheses.

$f (- \frac{\sqrt[4]{125}}{5}) = - {(- \frac{\sqrt[4]{125}}{5})}^{5} - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2

Simplify each term.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

Apply the product rule to $- \frac{\sqrt[4]{125}}{5}$ .

$f (- \frac{\sqrt[4]{125}}{5}) = - ({(- 1)}^{5} {(\frac{\sqrt[4]{125}}{5})}^{5}) - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.1.2

Apply the product rule to $\frac{\sqrt[4]{125}}{5}$ .

$f (- \frac{\sqrt[4]{125}}{5}) = - ({(- 1)}^{5} (\frac{{\sqrt[4]{125}}^{5}}{5^{5}})) - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.2

Multiply $- 1$ by ${(- 1)}^{5}$ by adding the exponents.

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Step 6.2.2.2.1

Move ${(- 1)}^{5}$ .

$f (- \frac{\sqrt[4]{125}}{5}) = {(- 1)}^{5} \cdot (- 1 \frac{{\sqrt[4]{125}}^{5}}{5^{5}}) - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.2.2

Multiply ${(- 1)}^{5}$ by $- 1$ .

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Step 6.2.2.2.2.1

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

$f (- \frac{\sqrt[4]{125}}{5}) = {(- 1)}^{5} \cdot ((- 1) (\frac{{\sqrt[4]{125}}^{5}}{5^{5}})) - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.2.2.2

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

$f (- \frac{\sqrt[4]{125}}{5}) = {(- 1)}^{5 + 1} (\frac{{\sqrt[4]{125}}^{5}}{5^{5}}) - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.2.3

Add $5$ and $1$ .

$f (- \frac{\sqrt[4]{125}}{5}) = {(- 1)}^{6} (\frac{{\sqrt[4]{125}}^{5}}{5^{5}}) - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.3

Raise $- 1$ to the power of $6$ .

$f (- \frac{\sqrt[4]{125}}{5}) = 1 (\frac{{\sqrt[4]{125}}^{5}}{5^{5}}) - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.4

Multiply $\frac{{\sqrt[4]{125}}^{5}}{5^{5}}$ by $1$ .

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{{\sqrt[4]{125}}^{5}}{5^{5}} - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.5

Simplify the numerator.

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Step 6.2.2.5.1

Rewrite ${\sqrt[4]{125}}^{5}$ as $\sqrt[4]{125^{5}}$ .

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{\sqrt[4]{125^{5}}}{5^{5}} - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.5.2

Raise $125$ to the power of $5$ .

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{\sqrt[4]{30517578125}}{5^{5}} - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.5.3

Rewrite $30517578125$ as $125^{4} \cdot 125$ .

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Step 6.2.2.5.3.1

Factor $244140625$ out of $30517578125$ .

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{\sqrt[4]{244140625 (125)}}{5^{5}} - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.5.3.2

Rewrite $244140625$ as $125^{4}$ .

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{\sqrt[4]{125^{4} \cdot 125}}{5^{5}} - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.5.4

Pull terms out from under the radical.

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{125 \sqrt[4]{125}}{5^{5}} - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.6

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

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{125 \sqrt[4]{125}}{3125} - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.7

Cancel the common factor of $125$ and $3125$ .

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Step 6.2.2.7.1

Factor $125$ out of $125 \sqrt[4]{125}$ .

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{125 (\sqrt[4]{125})}{3125} - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.7.2

Cancel the common factors.

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Step 6.2.2.7.2.1

Factor $125$ out of $3125$ .

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{125 \sqrt[4]{125}}{125 \cdot 25} - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.7.2.2

Cancel the common factor.

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{125 \sqrt[4]{125}}{125 \cdot 25} - \frac{\sqrt[4]{125}}{5}$

Step 6.2.2.7.2.3

Rewrite the expression.

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{\sqrt[4]{125}}{25} - \frac{\sqrt[4]{125}}{5}$

Step 6.2.3

To write $- \frac{\sqrt[4]{125}}{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{\sqrt[4]{125}}{25} - \frac{\sqrt[4]{125}}{5} \cdot \frac{5}{5}$

Step 6.2.4

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.4.1

Multiply $\frac{\sqrt[4]{125}}{5}$ by $\frac{5}{5}$ .

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{\sqrt[4]{125}}{25} - \frac{\sqrt[4]{125} \cdot 5}{5 \cdot 5}$

Step 6.2.4.2

Multiply $5$ by $5$ .

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{\sqrt[4]{125}}{25} - \frac{\sqrt[4]{125} \cdot 5}{25}$

Step 6.2.5

Combine the numerators over the common denominator.

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{\sqrt[4]{125} - \sqrt[4]{125} \cdot 5}{25}$

Step 6.2.6

Simplify the numerator.

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Step 6.2.6.1

Multiply $5$ by $- 1$ .

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{\sqrt[4]{125} - 5 \sqrt[4]{125}}{25}$

Step 6.2.6.2

Subtract $5 \sqrt[4]{125}$ from $\sqrt[4]{125}$ .

$f (- \frac{\sqrt[4]{125}}{5}) = \frac{- 4 \sqrt[4]{125}}{25}$

Step 6.2.7

Move the negative in front of the fraction.

$f (- \frac{\sqrt[4]{125}}{5}) = - \frac{4 \sqrt[4]{125}}{25}$

Step 6.2.8

The final answer is $- \frac{4 \sqrt[4]{125}}{25}$ .

$- \frac{4 \sqrt[4]{125}}{25}$

Step 7

The horizontal tangent lines on function $f (x) = - x^{5} + x$ are $y = \frac{4 \sqrt[4]{125}}{25}, y = - \frac{4 \sqrt[4]{125}}{25}$ .

$y = \frac{4 \sqrt[4]{125}}{25}, y = - \frac{4 \sqrt[4]{125}}{25}$

Step 8