Calculus Examples

$y = \sqrt[4]{x}$

Step 1

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

$f (x) = \sqrt[4]{x}$

Step 2

Find the derivative.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$\frac{1}{4} x^{\frac{1}{4} - 1 \cdot \frac{4}{4}}$

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{4} x^{\frac{1 - 1 \cdot 4}{4}}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $4$ .

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

Step 2.6.2

Subtract $4$ from $1$ .

$\frac{1}{4} x^{\frac{- 3}{4}}$

Step 2.7

Move the negative in front of the fraction.

$\frac{1}{4} x^{- \frac{3}{4}}$

Step 2.8

Simplify.

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Step 2.8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{4} \cdot \frac{1}{x^{\frac{3}{4}}}$

Step 2.8.2

Multiply $\frac{1}{4}$ by $\frac{1}{x^{\frac{3}{4}}}$ .

$\frac{1}{4 x^{\frac{3}{4}}}$

Step 3

Set the derivative equal to $0$ then solve the equation $\frac{1}{4 x^{\frac{3}{4}}} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$1 = 0$

Step 3.2

Since $1 \neq 0$ , there are no solutions.

No solution

Step 4

There are no solution found by setting the derivative equal to $0$ , $\frac{1}{4 x^{\frac{3}{4}}} = 0$ so there are no horizontal tangent lines.

No horizontal tangent lines found

Step 5