Calculus Examples

$y = \frac{1}{\sqrt[4]{9 x^{2} - 5}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{9 x^{2} - 5}$ as ${(9 x^{2} - 5)}^{\frac{1}{4}}$ .

$y = \frac{1}{{(9 x^{2} - 5)}^{\frac{1}{4}}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1}{{(9 x^{2} - 5)}^{\frac{1}{4}}})$

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

Apply basic rules of exponents.

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Step 4.1.1

Rewrite $\frac{1}{{(9 x^{2} - 5)}^{\frac{1}{4}}}$ as ${({(9 x^{2} - 5)}^{\frac{1}{4}})}^{- 1}$ .

$\frac{d}{d x} [{({(9 x^{2} - 5)}^{\frac{1}{4}})}^{- 1}]$

Step 4.1.2

Multiply the exponents in ${({(9 x^{2} - 5)}^{\frac{1}{4}})}^{- 1}$ .

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Step 4.1.2.1

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

$\frac{d}{d x} [{(9 x^{2} - 5)}^{\frac{1}{4} \cdot - 1}]$

Step 4.1.2.2

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

$\frac{d}{d x} [{(9 x^{2} - 5)}^{\frac{- 1}{4}}]$

Step 4.1.2.3

Move the negative in front of the fraction.

$\frac{d}{d x} [{(9 x^{2} - 5)}^{- \frac{1}{4}}]$

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- \frac{1}{4}}$ and $g (x) = 9 x^{2} - 5$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $9 x^{2} - 5$ .

$\frac{d}{d u} [u^{- \frac{1}{4}}] \frac{d}{d x} [9 x^{2} - 5]$

Step 4.2.2

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

$- \frac{1}{4} u^{- \frac{1}{4} - 1} \frac{d}{d x} [9 x^{2} - 5]$

Step 4.2.3

Replace all occurrences of $u$ with $9 x^{2} - 5$ .

$- \frac{1}{4} {(9 x^{2} - 5)}^{- \frac{1}{4} - 1} \frac{d}{d x} [9 x^{2} - 5]$

Step 4.3

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

$- \frac{1}{4} {(9 x^{2} - 5)}^{- \frac{1}{4} - 1 \cdot \frac{4}{4}} \frac{d}{d x} [9 x^{2} - 5]$

Step 4.4

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

$- \frac{1}{4} {(9 x^{2} - 5)}^{- \frac{1}{4} + \frac{- 1 \cdot 4}{4}} \frac{d}{d x} [9 x^{2} - 5]$

Step 4.5

Combine the numerators over the common denominator.

$- \frac{1}{4} {(9 x^{2} - 5)}^{\frac{- 1 - 1 \cdot 4}{4}} \frac{d}{d x} [9 x^{2} - 5]$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $4$ .

$- \frac{1}{4} {(9 x^{2} - 5)}^{\frac{- 1 - 4}{4}} \frac{d}{d x} [9 x^{2} - 5]$

Step 4.6.2

Subtract $4$ from $- 1$ .

$- \frac{1}{4} {(9 x^{2} - 5)}^{\frac{- 5}{4}} \frac{d}{d x} [9 x^{2} - 5]$

Step 4.7

Combine fractions.

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Step 4.7.1

Move the negative in front of the fraction.

$- \frac{1}{4} {(9 x^{2} - 5)}^{- \frac{5}{4}} \frac{d}{d x} [9 x^{2} - 5]$

Step 4.7.2

Combine ${(9 x^{2} - 5)}^{- \frac{5}{4}}$ and $\frac{1}{4}$ .

$- \frac{{(9 x^{2} - 5)}^{- \frac{5}{4}}}{4} \frac{d}{d x} [9 x^{2} - 5]$

Step 4.7.3

Move ${(9 x^{2} - 5)}^{- \frac{5}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}} \frac{d}{d x} [9 x^{2} - 5]$

Step 4.8

By the Sum Rule, the derivative of $9 x^{2} - 5$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- 5]$ .

$- \frac{1}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}} (\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- 5])$

Step 4.9

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$- \frac{1}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}} (9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5])$

Step 4.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- \frac{1}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}} (9 (2 x) + \frac{d}{d x} [- 5])$

Step 4.11

Multiply $2$ by $9$ .

$- \frac{1}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}} (18 x + \frac{d}{d x} [- 5])$

Step 4.12

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$- \frac{1}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}} (18 x + 0)$

Step 4.13

Simplify terms.

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Step 4.13.1

Add $18 x$ and $0$ .

$- \frac{1}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}} (18 x)$

Step 4.13.2

Multiply $18$ by $- 1$ .

$- 18 \frac{1}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}} x$

Step 4.13.3

Combine $- 18$ and $\frac{1}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}}$ .

$\frac{- 18}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}} x$

Step 4.13.4

Combine $\frac{- 18}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}}$ and $x$ .

$\frac{- 18 x}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}}$

Step 4.13.5

Factor $2$ out of $- 18 x$ .

$\frac{2 (- 9 x)}{4 {(9 x^{2} - 5)}^{\frac{5}{4}}}$

Step 4.14

Cancel the common factors.

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Step 4.14.1

Factor $2$ out of $4 {(9 x^{2} - 5)}^{\frac{5}{4}}$ .

$\frac{2 (- 9 x)}{2 (2 {(9 x^{2} - 5)}^{\frac{5}{4}})}$

Step 4.14.2

Cancel the common factor.

$\frac{2 (- 9 x)}{2 (2 {(9 x^{2} - 5)}^{\frac{5}{4}})}$

Step 4.14.3

Rewrite the expression.

$\frac{- 9 x}{2 {(9 x^{2} - 5)}^{\frac{5}{4}}}$

Step 4.15

Move the negative in front of the fraction.

$- \frac{9 x}{2 {(9 x^{2} - 5)}^{\frac{5}{4}}}$

Step 5

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

$y' = - \frac{9 x}{2 {(9 x^{2} - 5)}^{\frac{5}{4}}}$

Step 6

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

$\frac{d y}{d x} = - \frac{9 x}{2 {(9 x^{2} - 5)}^{\frac{5}{4}}}$