Calculus Examples

$y = \frac{2}{\sqrt[6]{5 x^{2} - 1}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{5 x^{2} - 1}$ as ${(5 x^{2} - 1)}^{\frac{1}{6}}$ .

$y = \frac{2}{{(5 x^{2} - 1)}^{\frac{1}{6}}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{2}{{(5 x^{2} - 1)}^{\frac{1}{6}}})$

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 Constant Multiple Rule.

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Step 4.1.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{{(5 x^{2} - 1)}^{\frac{1}{6}}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{{(5 x^{2} - 1)}^{\frac{1}{6}}}]$ .

$2 \frac{d}{d x} [\frac{1}{{(5 x^{2} - 1)}^{\frac{1}{6}}}]$

Step 4.1.2

Apply basic rules of exponents.

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Step 4.1.2.1

Rewrite $\frac{1}{{(5 x^{2} - 1)}^{\frac{1}{6}}}$ as ${({(5 x^{2} - 1)}^{\frac{1}{6}})}^{- 1}$ .

$2 \frac{d}{d x} [{({(5 x^{2} - 1)}^{\frac{1}{6}})}^{- 1}]$

Step 4.1.2.2

Multiply the exponents in ${({(5 x^{2} - 1)}^{\frac{1}{6}})}^{- 1}$ .

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Step 4.1.2.2.1

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

$2 \frac{d}{d x} [{(5 x^{2} - 1)}^{\frac{1}{6} \cdot - 1}]$

Step 4.1.2.2.2

Combine $\frac{1}{6}$ and $- 1$ .

$2 \frac{d}{d x} [{(5 x^{2} - 1)}^{\frac{- 1}{6}}]$

Step 4.1.2.2.3

Move the negative in front of the fraction.

$2 \frac{d}{d x} [{(5 x^{2} - 1)}^{- \frac{1}{6}}]$

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 $f (x) = x^{- \frac{1}{6}}$ and $g (x) = 5 x^{2} - 1$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $5 x^{2} - 1$ .

$2 (\frac{d}{d u} [u^{- \frac{1}{6}}] \frac{d}{d x} [5 x^{2} - 1])$

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - \frac{1}{6}$ .

$2 (- \frac{1}{6} u^{- \frac{1}{6} - 1} \frac{d}{d x} [5 x^{2} - 1])$

Step 4.2.3

Replace all occurrences of $u$ with $5 x^{2} - 1$ .

$2 (- \frac{1}{6} {(5 x^{2} - 1)}^{- \frac{1}{6} - 1} \frac{d}{d x} [5 x^{2} - 1])$

Step 4.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 (- \frac{1}{6} {(5 x^{2} - 1)}^{- \frac{1}{6} - 1 \cdot \frac{6}{6}} \frac{d}{d x} [5 x^{2} - 1])$

Step 4.4

Combine $- 1$ and $\frac{6}{6}$ .

$2 (- \frac{1}{6} {(5 x^{2} - 1)}^{- \frac{1}{6} + \frac{- 1 \cdot 6}{6}} \frac{d}{d x} [5 x^{2} - 1])$

Step 4.5

Combine the numerators over the common denominator.

$2 (- \frac{1}{6} {(5 x^{2} - 1)}^{\frac{- 1 - 1 \cdot 6}{6}} \frac{d}{d x} [5 x^{2} - 1])$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $6$ .

$2 (- \frac{1}{6} {(5 x^{2} - 1)}^{\frac{- 1 - 6}{6}} \frac{d}{d x} [5 x^{2} - 1])$

Step 4.6.2

Subtract $6$ from $- 1$ .

$2 (- \frac{1}{6} {(5 x^{2} - 1)}^{\frac{- 7}{6}} \frac{d}{d x} [5 x^{2} - 1])$

Step 4.7

Move the negative in front of the fraction.

$2 (- \frac{1}{6} {(5 x^{2} - 1)}^{- \frac{7}{6}} \frac{d}{d x} [5 x^{2} - 1])$

Step 4.8

Combine ${(5 x^{2} - 1)}^{- \frac{7}{6}}$ and $\frac{1}{6}$ .

$2 (- \frac{{(5 x^{2} - 1)}^{- \frac{7}{6}}}{6} \frac{d}{d x} [5 x^{2} - 1])$

Step 4.9

Simplify the expression.

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Step 4.9.1

Move ${(5 x^{2} - 1)}^{- \frac{7}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 (- \frac{1}{6 {(5 x^{2} - 1)}^{\frac{7}{6}}} \frac{d}{d x} [5 x^{2} - 1])$

Step 4.9.2

Multiply $- 1$ by $2$ .

$- 2 (\frac{1}{6 {(5 x^{2} - 1)}^{\frac{7}{6}}} \frac{d}{d x} [5 x^{2} - 1])$

Step 4.10

Combine $\frac{1}{6 {(5 x^{2} - 1)}^{\frac{7}{6}}}$ and $- 2$ .

$\frac{- 2}{6 {(5 x^{2} - 1)}^{\frac{7}{6}}} \frac{d}{d x} [5 x^{2} - 1]$

Step 4.11

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{6 {(5 x^{2} - 1)}^{\frac{7}{6}}} \frac{d}{d x} [5 x^{2} - 1]$

Step 4.12

Cancel the common factors.

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Step 4.12.1

Factor $2$ out of $6 {(5 x^{2} - 1)}^{\frac{7}{6}}$ .

$\frac{2 (- 1)}{2 (3 {(5 x^{2} - 1)}^{\frac{7}{6}})} \frac{d}{d x} [5 x^{2} - 1]$

Step 4.12.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 (3 {(5 x^{2} - 1)}^{\frac{7}{6}})} \frac{d}{d x} [5 x^{2} - 1]$

Step 4.12.3

Rewrite the expression.

$\frac{- 1}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}} \frac{d}{d x} [5 x^{2} - 1]$

Step 4.13

Move the negative in front of the fraction.

$- \frac{1}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}} \frac{d}{d x} [5 x^{2} - 1]$

Step 4.14

By the Sum Rule, the derivative of $5 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 1]$ .

$- \frac{1}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}} (\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 1])$

Step 4.15

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$- \frac{1}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}} (5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])$

Step 4.16

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

$- \frac{1}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}} (5 (2 x) + \frac{d}{d x} [- 1])$

Step 4.17

Multiply $2$ by $5$ .

$- \frac{1}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}} (10 x + \frac{d}{d x} [- 1])$

Step 4.18

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

$- \frac{1}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}} (10 x + 0)$

Step 4.19

Combine fractions.

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Step 4.19.1

Add $10 x$ and $0$ .

$- \frac{1}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}} (10 x)$

Step 4.19.2

Multiply $10$ by $- 1$ .

$- 10 \frac{1}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}} x$

Step 4.19.3

Combine $- 10$ and $\frac{1}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}}$ .

$\frac{- 10}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}} x$

Step 4.19.4

Combine $\frac{- 10}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}}$ and $x$ .

$\frac{- 10 x}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}}$

Step 4.19.5

Move the negative in front of the fraction.

$- \frac{10 x}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}}$

Step 5

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

$y' = - \frac{10 x}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}}$

Step 6

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

$\frac{d y}{d x} = - \frac{10 x}{3 {(5 x^{2} - 1)}^{\frac{7}{6}}}$