Calculus Examples

$y = - \frac{1}{\sqrt[4]{x}}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (- \frac{1}{x^{\frac{1}{4}}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{\frac{1}{4}}}$ .

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

Step 4.2

Differentiate using the Power Rule.

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Step 4.2.1

Rewrite $\frac{1}{x^{\frac{1}{4}}}$ as ${(x^{\frac{1}{4}})}^{- 1}$ .

$- \frac{d}{d x} [{(x^{\frac{1}{4}})}^{- 1}] + \frac{1}{x^{\frac{1}{4}}} \frac{d}{d x} [- 1]$

Step 4.2.2

Multiply the exponents in ${(x^{\frac{1}{4}})}^{- 1}$ .

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

Move the negative in front of the fraction.

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

Step 4.2.3

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

$- (- \frac{1}{4} x^{- \frac{1}{4} - 1}) + \frac{1}{x^{\frac{1}{4}}} \frac{d}{d x} [- 1]$

Step 4.3

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

$- (- \frac{1}{4} x^{- \frac{1}{4} - 1 \cdot \frac{4}{4}}) + \frac{1}{x^{\frac{1}{4}}} \frac{d}{d x} [- 1]$

Step 4.4

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

$- (- \frac{1}{4} x^{- \frac{1}{4} + \frac{- 1 \cdot 4}{4}}) + \frac{1}{x^{\frac{1}{4}}} \frac{d}{d x} [- 1]$

Step 4.5

Combine the numerators over the common denominator.

$- (- \frac{1}{4} x^{\frac{- 1 - 1 \cdot 4}{4}}) + \frac{1}{x^{\frac{1}{4}}} \frac{d}{d x} [- 1]$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $4$ .

$- (- \frac{1}{4} x^{\frac{- 1 - 4}{4}}) + \frac{1}{x^{\frac{1}{4}}} \frac{d}{d x} [- 1]$

Step 4.6.2

Subtract $4$ from $- 1$ .

$- (- \frac{1}{4} x^{\frac{- 5}{4}}) + \frac{1}{x^{\frac{1}{4}}} \frac{d}{d x} [- 1]$

Step 4.7

Combine fractions.

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Step 4.7.1

Move the negative in front of the fraction.

$- (- \frac{1}{4} x^{- \frac{5}{4}}) + \frac{1}{x^{\frac{1}{4}}} \frac{d}{d x} [- 1]$

Step 4.7.2

Combine $x^{- \frac{5}{4}}$ and $\frac{1}{4}$ .

$- - \frac{x^{- \frac{5}{4}}}{4} + \frac{1}{x^{\frac{1}{4}}} \frac{d}{d x} [- 1]$

Step 4.7.3

Simplify the expression.

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Step 4.7.3.1

Multiply $- 1$ by $- 1$ .

$1 \frac{x^{- \frac{5}{4}}}{4} + \frac{1}{x^{\frac{1}{4}}} \frac{d}{d x} [- 1]$

Step 4.7.3.2

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

$\frac{x^{- \frac{5}{4}}}{4} + \frac{1}{x^{\frac{1}{4}}} \frac{d}{d x} [- 1]$

Step 4.7.3.3

Move $x^{- \frac{5}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{4 x^{\frac{5}{4}}} + \frac{1}{x^{\frac{1}{4}}} \frac{d}{d x} [- 1]$

Step 4.8

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{1}{4 x^{\frac{5}{4}}} + \frac{1}{x^{\frac{1}{4}}} \cdot 0$

Step 4.9

Simplify the expression.

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Step 4.9.1

Multiply $\frac{1}{x^{\frac{1}{4}}}$ by $0$ .

$\frac{1}{4 x^{\frac{5}{4}}} + 0$

Step 4.9.2

Add $\frac{1}{4 x^{\frac{5}{4}}}$ and $0$ .

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

Step 5

Reform the equation by setting the left side equal to the right side.

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

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

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