Calculus Examples

$y = - 100 x^{\frac{1}{4}}$

Step 1

Write $y = - 100 x^{\frac{1}{4}}$ as a function.

$f (x) = - 100 x^{\frac{1}{4}}$

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Since $- 100$ is constant with respect to $x$ , the derivative of $- 100 x^{\frac{1}{4}}$ with respect to $x$ is $- 100 \frac{d}{d x} [x^{\frac{1}{4}}]$ .

$- 100 \frac{d}{d x} [x^{\frac{1}{4}}]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{4}$ .

$- 100 (\frac{1}{4} x^{\frac{1}{4} - 1})$

Step 2.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- 100 (\frac{1}{4} x^{\frac{1}{4} - 1 \cdot \frac{4}{4}})$

Step 2.1.4

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

$- 100 (\frac{1}{4} x^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}})$

Step 2.1.5

Combine the numerators over the common denominator.

$- 100 (\frac{1}{4} x^{\frac{1 - 1 \cdot 4}{4}})$

Step 2.1.6

Simplify the numerator.

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Step 2.1.6.1

Multiply $- 1$ by $4$ .

$- 100 (\frac{1}{4} x^{\frac{1 - 4}{4}})$

Step 2.1.6.2

Subtract $4$ from $1$ .

$- 100 (\frac{1}{4} x^{\frac{- 3}{4}})$

Step 2.1.7

Move the negative in front of the fraction.

$- 100 (\frac{1}{4} x^{- \frac{3}{4}})$

Step 2.1.8

Combine $\frac{1}{4}$ and $x^{- \frac{3}{4}}$ .

$- 100 \frac{x^{- \frac{3}{4}}}{4}$

Step 2.1.9

Combine $- 100$ and $\frac{x^{- \frac{3}{4}}}{4}$ .

$\frac{- 100 x^{- \frac{3}{4}}}{4}$

Step 2.1.10

Move $x^{- \frac{3}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 100}{4 x^{\frac{3}{4}}}$

Step 2.1.11

Factor $4$ out of $- 100$ .

$\frac{4 \cdot - 25}{4 x^{\frac{3}{4}}}$

Step 2.1.12

Cancel the common factors.

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Step 2.1.12.1

Factor $4$ out of $4 x^{\frac{3}{4}}$ .

$\frac{4 \cdot - 25}{4 (x^{\frac{3}{4}})}$

Step 2.1.12.2

Cancel the common factor.

$\frac{4 \cdot - 25}{4 x^{\frac{3}{4}}}$

Step 2.1.12.3

Rewrite the expression.

$\frac{- 25}{x^{\frac{3}{4}}}$

Step 2.1.13

Move the negative in front of the fraction.

$f' (x) = - \frac{25}{x^{\frac{3}{4}}}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{25}{x^{\frac{3}{4}}}$ .

$- \frac{25}{x^{\frac{3}{4}}}$

Step 3

Set the first derivative equal to $0$ then solve the equation $- \frac{25}{x^{\frac{3}{4}}} = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$- \frac{25}{x^{\frac{3}{4}}} = 0$

Step 3.2

Set the numerator equal to zero.

$25 = 0$

Step 3.3

Since $25 \neq 0$ , there are no solutions.

No solution

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

Step 5

Find where the derivative is undefined.

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Step 5.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- \frac{25}{\sqrt[4]{x^{3}}}$

Step 5.2

Set the denominator in $\frac{25}{\sqrt[4]{x^{3}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[4]{x^{3}} = 0$

Step 5.3

Solve for $x$ .

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Step 5.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{x^{3}}}^{4} = 0^{4}$

Step 5.3.2

Simplify each side of the equation.

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Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

${(x^{\frac{3}{4}})}^{4} = 0^{4}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Multiply the exponents in ${(x^{\frac{3}{4}})}^{4}$ .

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Step 5.3.2.2.1.1

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

$x^{\frac{3}{4} \cdot 4} = 0^{4}$

Step 5.3.2.2.1.2

Cancel the common factor of $4$ .

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Step 5.3.2.2.1.2.1

Cancel the common factor.

$x^{\frac{3}{4} \cdot 4} = 0^{4}$

Step 5.3.2.2.1.2.2

Rewrite the expression.

$x^{3} = 0^{4}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$x^{3} = 0$

Step 5.3.3

Solve for $x$ .

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Step 5.3.3.1

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

$x = \sqrt[3]{0}$

Step 5.3.3.2

Simplify $\sqrt[3]{0}$ .

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Step 5.3.3.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 5.3.3.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 5.4

Set the radicand in $\sqrt[4]{x^{3}}$ less than $0$ to find where the expression is undefined.

$x^{3} < 0$

Step 5.5

Solve for $x$ .

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Step 5.5.1

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

$\sqrt[3]{x^{3}} < \sqrt[3]{0}$

Step 5.5.2

Simplify the equation.

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Step 5.5.2.1

Simplify the left side.

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Step 5.5.2.1.1

Pull terms out from under the radical.

$x < \sqrt[3]{0}$

Step 5.5.2.2

Simplify the right side.

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Step 5.5.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 5.5.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x < \sqrt[3]{0^{3}}$

Step 5.5.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 5.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 6

After finding the point that makes the derivative $f' (x) = - \frac{25}{x^{\frac{3}{4}}}$ equal to $0$ or undefined, the interval to check where $f (x) = - 100 x^{\frac{1}{4}}$ is increasing and where it is decreasing is $(- \infty, 0) \cup (0, \infty)$ .

$(- \infty, 0) \cup (0, \infty)$

Step 7

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = - \frac{25}{{(- 1)}^{\frac{3}{4}}}$

Step 7.2

The final answer is $- \frac{25}{{(- 1)}^{\frac{3}{4}}}$ .

$- \frac{25}{{(- 1)}^{\frac{3}{4}}}$

Step 7.3

At $x = - 1$ the derivative is $- \frac{25}{{(- 1)}^{\frac{3}{4}}}$ . Since this is negative, the function is decreasing on $(- \infty, 0)$ .

Decreasing on $(- \infty, 0)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(0, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = - \frac{25}{{(1)}^{\frac{3}{4}}}$

Step 8.2

Simplify the result.

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Step 8.2.1

One to any power is one.

$f' (1) = - \frac{25}{1}$

Step 8.2.2

Divide $25$ by $1$ .

$f' (1) = - 1 \cdot 25$

Step 8.2.3

Multiply $- 1$ by $25$ .

$f' (1) = - 25$

Step 8.2.4

The final answer is $- 25$ .

$- 25$

Step 8.3

At $x = 1$ the derivative is $- 25$ . Since this is negative, the function is decreasing on $(0, \infty)$ .

Decreasing on $(0, \infty)$ since $f' (x) < 0$

Step 9

List the intervals on which the function is increasing and decreasing.

Decreasing on: $(- \infty, 0), (0, \infty)$

Step 10