Calculus Examples

$y = \ln (\frac{5 x - 4}{x \sqrt[7]{x^{2} + 1}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x^{2} + 1}$ as ${(x^{2} + 1)}^{\frac{1}{7}}$ .

$y = \ln (\frac{5 x - 4}{x {(x^{2} + 1)}^{\frac{1}{7}}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (\frac{5 x - 4}{x {(x^{2} + 1)}^{\frac{1}{7}}}))$

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 chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{5 x - 4}{x {(x^{2} + 1)}^{\frac{1}{7}}}$ .

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Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{5 x - 4}{x {(x^{2} + 1)}^{\frac{1}{7}}}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\frac{5 x - 4}{x {(x^{2} + 1)}^{\frac{1}{7}}}]$

Step 4.1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{1}{u_{1}} \frac{d}{d x} [\frac{5 x - 4}{x {(x^{2} + 1)}^{\frac{1}{7}}}]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with $\frac{5 x - 4}{x {(x^{2} + 1)}^{\frac{1}{7}}}$ .

$\frac{1}{\frac{5 x - 4}{x {(x^{2} + 1)}^{\frac{1}{7}}}} \frac{d}{d x} [\frac{5 x - 4}{x {(x^{2} + 1)}^{\frac{1}{7}}}]$

Step 4.2

Multiply by the reciprocal of the fraction to divide by $\frac{5 x - 4}{x {(x^{2} + 1)}^{\frac{1}{7}}}$ .

$1 \frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \frac{d}{d x} [\frac{5 x - 4}{x {(x^{2} + 1)}^{\frac{1}{7}}}]$

Step 4.3

Multiply $\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4}$ by $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \frac{d}{d x} [\frac{5 x - 4}{x {(x^{2} + 1)}^{\frac{1}{7}}}]$

Step 4.4

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 5 x - 4$ and $g (x) = x {(x^{2} + 1)}^{\frac{1}{7}}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [5 x - 4] - (5 x - 4) \frac{d}{d x} [x {(x^{2} + 1)}^{\frac{1}{7}}]}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.5

Differentiate.

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Step 4.5.1

By the Sum Rule, the derivative of $5 x - 4$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 4]$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{x {(x^{2} + 1)}^{\frac{1}{7}} (\frac{d}{d x} [5 x] + \frac{d}{d x} [- 4]) - (5 x - 4) \frac{d}{d x} [x {(x^{2} + 1)}^{\frac{1}{7}}]}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.5.2

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

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 \frac{d}{d x} [x] + \frac{d}{d x} [- 4]) - (5 x - 4) \frac{d}{d x} [x {(x^{2} + 1)}^{\frac{1}{7}}]}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.5.3

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

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 \cdot 1 + \frac{d}{d x} [- 4]) - (5 x - 4) \frac{d}{d x} [x {(x^{2} + 1)}^{\frac{1}{7}}]}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.5.4

Multiply $5$ by $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 + \frac{d}{d x} [- 4]) - (5 x - 4) \frac{d}{d x} [x {(x^{2} + 1)}^{\frac{1}{7}}]}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.5.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 + 0) - (5 x - 4) \frac{d}{d x} [x {(x^{2} + 1)}^{\frac{1}{7}}]}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.5.6

Simplify the expression.

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Step 4.5.6.1

Add $5$ and $0$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{x {(x^{2} + 1)}^{\frac{1}{7}} \cdot 5 - (5 x - 4) \frac{d}{d x} [x {(x^{2} + 1)}^{\frac{1}{7}}]}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.5.6.2

Move $5$ to the left of $x {(x^{2} + 1)}^{\frac{1}{7}}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 \cdot (x {(x^{2} + 1)}^{\frac{1}{7}}) - (5 x - 4) \frac{d}{d x} [x {(x^{2} + 1)}^{\frac{1}{7}}]}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.6

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 $f (x) = x$ and $g (x) = {(x^{2} + 1)}^{\frac{1}{7}}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (x \frac{d}{d x} [{(x^{2} + 1)}^{\frac{1}{7}}] + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{7}}$ and $g (x) = x^{2} + 1$ .

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Step 4.7.1

To apply the Chain Rule, set $u_{2}$ as $x^{2} + 1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (x (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{7}}] \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.7.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{1}{7}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (x (\frac{1}{7} {u_{2}}^{\frac{1}{7} - 1} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.7.3

Replace all occurrences of $u_{2}$ with $x^{2} + 1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (x (\frac{1}{7} {(x^{2} + 1)}^{\frac{1}{7} - 1} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (x (\frac{1}{7} {(x^{2} + 1)}^{\frac{1}{7} - 1 \cdot \frac{7}{7}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.9

Combine $- 1$ and $\frac{7}{7}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (x (\frac{1}{7} {(x^{2} + 1)}^{\frac{1}{7} + \frac{- 1 \cdot 7}{7}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.10

Combine the numerators over the common denominator.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (x (\frac{1}{7} {(x^{2} + 1)}^{\frac{1 - 1 \cdot 7}{7}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.11

Simplify the numerator.

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Step 4.11.1

Multiply $- 1$ by $7$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (x (\frac{1}{7} {(x^{2} + 1)}^{\frac{1 - 7}{7}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.11.2

Subtract $7$ from $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (x (\frac{1}{7} {(x^{2} + 1)}^{\frac{- 6}{7}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.12

Combine fractions.

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Step 4.12.1

Move the negative in front of the fraction.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (x (\frac{1}{7} {(x^{2} + 1)}^{- \frac{6}{7}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.12.2

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

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (x (\frac{{(x^{2} + 1)}^{- \frac{6}{7}}}{7} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.12.3

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

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (x (\frac{1}{7 {(x^{2} + 1)}^{\frac{6}{7}}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.12.4

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

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{x}{7 {(x^{2} + 1)}^{\frac{6}{7}}} \frac{d}{d x} [x^{2} + 1] + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.13

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

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{x}{7 {(x^{2} + 1)}^{\frac{6}{7}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.14

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

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{x}{7 {(x^{2} + 1)}^{\frac{6}{7}}} (2 x + \frac{d}{d x} [1]) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.15

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

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{x}{7 {(x^{2} + 1)}^{\frac{6}{7}}} (2 x + 0) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.16

Combine fractions.

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Step 4.16.1

Add $2 x$ and $0$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{x}{7 {(x^{2} + 1)}^{\frac{6}{7}}} (2 x) + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.16.2

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

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{2 x}{7 {(x^{2} + 1)}^{\frac{6}{7}}} x + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.16.3

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

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{2 x \cdot x}{7 {(x^{2} + 1)}^{\frac{6}{7}}} + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.17

Raise $x$ to the power of $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{2 (x^{1} x)}{7 {(x^{2} + 1)}^{\frac{6}{7}}} + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.18

Raise $x$ to the power of $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{2 (x^{1} x^{1})}{7 {(x^{2} + 1)}^{\frac{6}{7}}} + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.19

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{2 x^{1 + 1}}{7 {(x^{2} + 1)}^{\frac{6}{7}}} + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.20

Add $1$ and $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{2 x^{2}}{7 {(x^{2} + 1)}^{\frac{6}{7}}} + {(x^{2} + 1)}^{\frac{1}{7}} \frac{d}{d x} [x])}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.21

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

Step 4.22

Multiply ${(x^{2} + 1)}^{\frac{1}{7}}$ by $1$ .

Step 4.23

To write ${(x^{2} + 1)}^{\frac{1}{7}}$ as a fraction with a common denominator, multiply by $\frac{7 {(x^{2} + 1)}^{\frac{6}{7}}}{7 {(x^{2} + 1)}^{\frac{6}{7}}}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{2 x^{2}}{7 {(x^{2} + 1)}^{\frac{6}{7}}} + {(x^{2} + 1)}^{\frac{1}{7}} \cdot \frac{7 {(x^{2} + 1)}^{\frac{6}{7}}}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.24

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

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) (\frac{2 x^{2}}{7 {(x^{2} + 1)}^{\frac{6}{7}}} + \frac{{(x^{2} + 1)}^{\frac{1}{7}} (7 {(x^{2} + 1)}^{\frac{6}{7}})}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.25

Combine the numerators over the common denominator.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) \frac{2 x^{2} + {(x^{2} + 1)}^{\frac{1}{7}} (7 {(x^{2} + 1)}^{\frac{6}{7}})}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.26

Multiply ${(x^{2} + 1)}^{\frac{1}{7}}$ by ${(x^{2} + 1)}^{\frac{6}{7}}$ by adding the exponents.

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Step 4.26.1

Move ${(x^{2} + 1)}^{\frac{6}{7}}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) \frac{2 x^{2} + {(x^{2} + 1)}^{\frac{6}{7}} {(x^{2} + 1)}^{\frac{1}{7}} \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.26.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) \frac{2 x^{2} + {(x^{2} + 1)}^{\frac{6}{7} + \frac{1}{7}} \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.26.3

Combine the numerators over the common denominator.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) \frac{2 x^{2} + {(x^{2} + 1)}^{\frac{6 + 1}{7}} \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.26.4

Add $6$ and $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) \frac{2 x^{2} + {(x^{2} + 1)}^{\frac{7}{7}} \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.26.5

Divide $7$ by $7$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) \frac{2 x^{2} + {(x^{2} + 1)}^{1} \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.27

Simplify the expression.

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Step 4.27.1

Simplify ${(x^{2} + 1)}^{1} \cdot 7$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) \frac{2 x^{2} + (x^{2} + 1) \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.27.2

Move $7$ to the left of $x^{2} + 1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4} \cdot \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) \frac{2 x^{2} + 7 \cdot (x^{2} + 1)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.28

Multiply $\frac{x {(x^{2} + 1)}^{\frac{1}{7}}}{5 x - 4}$ by $\frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) \frac{2 x^{2} + 7 (x^{2} + 1)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{{(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) \frac{2 x^{2} + 7 (x^{2} + 1)}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{(5 x - 4) {(x {(x^{2} + 1)}^{\frac{1}{7}})}^{2}}$

Step 4.29

Simplify.

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Step 4.29.1

Apply the product rule to $x {(x^{2} + 1)}^{\frac{1}{7}}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 x {(x^{2} + 1)}^{\frac{1}{7}} - (5 x - 4) \frac{2 x^{2} + 7 (x^{2} + 1)}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.2

Apply the distributive property.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 x {(x^{2} + 1)}^{\frac{1}{7}} + (- (5 x) - - 4) \frac{2 x^{2} + 7 (x^{2} + 1)}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.3

Apply the distributive property.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 x {(x^{2} + 1)}^{\frac{1}{7}} + (- (5 x) - - 4) \frac{2 x^{2} + 7 x^{2} + 7 \cdot 1}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4

Simplify the numerator.

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Step 4.29.4.1

Combine exponents.

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Step 4.29.4.1.1

Multiply $- 1$ by $5$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 x {(x^{2} + 1)}^{\frac{1}{7}} + (- 5 x - - 4) \frac{2 x^{2} + 7 x^{2} + 7 \cdot 1}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.1.2

Multiply $- 1$ by $- 4$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 x {(x^{2} + 1)}^{\frac{1}{7}} + (- 5 x + 4) \frac{2 x^{2} + 7 x^{2} + 7 \cdot 1}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.1.3

Multiply $7$ by $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 x {(x^{2} + 1)}^{\frac{1}{7}} + (- 5 x + 4) \frac{2 x^{2} + 7 x^{2} + 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.2

Simplify each term.

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Step 4.29.4.2.1

Add $2 x^{2}$ and $7 x^{2}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 x {(x^{2} + 1)}^{\frac{1}{7}} + (- 5 x + 4) \frac{9 x^{2} + 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.2.2

Multiply $- 5 x + 4$ by $\frac{9 x^{2} + 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 x {(x^{2} + 1)}^{\frac{1}{7}} + \frac{(- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.3

To write $5 x {(x^{2} + 1)}^{\frac{1}{7}}$ as a fraction with a common denominator, multiply by $\frac{7 {(x^{2} + 1)}^{\frac{6}{7}}}{7 {(x^{2} + 1)}^{\frac{6}{7}}}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} (5 x {(x^{2} + 1)}^{\frac{1}{7}} \cdot \frac{7 {(x^{2} + 1)}^{\frac{6}{7}}}{7 {(x^{2} + 1)}^{\frac{6}{7}}} + \frac{(- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.4

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

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} (\frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} (7 {(x^{2} + 1)}^{\frac{6}{7}})}{7 {(x^{2} + 1)}^{\frac{6}{7}}} + \frac{(- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}})}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.5

Combine the numerators over the common denominator.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{5 x {(x^{2} + 1)}^{\frac{1}{7}} (7 {(x^{2} + 1)}^{\frac{6}{7}}) + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6

Simplify the numerator.

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Step 4.29.4.6.1

Multiply ${(x^{2} + 1)}^{\frac{1}{7}}$ by ${(x^{2} + 1)}^{\frac{6}{7}}$ by adding the exponents.

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Step 4.29.4.6.1.1

Move ${(x^{2} + 1)}^{\frac{6}{7}}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{5 x ({(x^{2} + 1)}^{\frac{6}{7}} {(x^{2} + 1)}^{\frac{1}{7}}) \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{5 x {(x^{2} + 1)}^{\frac{6}{7} + \frac{1}{7}} \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.1.3

Combine the numerators over the common denominator.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{5 x {(x^{2} + 1)}^{\frac{6 + 1}{7}} \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.1.4

Add $6$ and $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{5 x {(x^{2} + 1)}^{\frac{7}{7}} \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.1.5

Divide $7$ by $7$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{5 x {(x^{2} + 1)}^{1} \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.2

Simplify $5 x {(x^{2} + 1)}^{1} \cdot 7$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{5 x (x^{2} + 1) \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.3

Apply the distributive property.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{(5 x \cdot x^{2} + 5 x \cdot 1) \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.4

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.29.4.6.4.1

Move $x^{2}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{(5 (x^{2} x) + 5 x \cdot 1) \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.4.2

Multiply $x^{2}$ by $x$ .

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Step 4.29.4.6.4.2.1

Raise $x$ to the power of $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{(5 (x^{2} x^{1}) + 5 x \cdot 1) \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{(5 x^{2 + 1} + 5 x \cdot 1) \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.4.3

Add $2$ and $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{(5 x^{3} + 5 x \cdot 1) \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.5

Multiply $5$ by $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{(5 x^{3} + 5 x) \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.6

Apply the distributive property.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{5 x^{3} \cdot 7 + 5 x \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.7

Multiply $7$ by $5$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 5 x \cdot 7 + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.8

Multiply $7$ by $5$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x + (- 5 x + 4) (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.9

Expand $(- 5 x + 4) (9 x^{2} + 7)$ using the FOIL Method.

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Step 4.29.4.6.9.1

Apply the distributive property.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x - 5 x (9 x^{2} + 7) + 4 (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.9.2

Apply the distributive property.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x - 5 x (9 x^{2}) - 5 x \cdot 7 + 4 (9 x^{2} + 7)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.9.3

Apply the distributive property.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x - 5 x (9 x^{2}) - 5 x \cdot 7 + 4 (9 x^{2}) + 4 \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.10

Simplify each term.

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Step 4.29.4.6.10.1

Rewrite using the commutative property of multiplication.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x - 5 \cdot 9 x \cdot x^{2} - 5 x \cdot 7 + 4 (9 x^{2}) + 4 \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.10.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.29.4.6.10.2.1

Move $x^{2}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x - 5 \cdot 9 (x^{2} x) - 5 x \cdot 7 + 4 (9 x^{2}) + 4 \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.10.2.2

Multiply $x^{2}$ by $x$ .

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Step 4.29.4.6.10.2.2.1

Raise $x$ to the power of $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x - 5 \cdot 9 (x^{2} x^{1}) - 5 x \cdot 7 + 4 (9 x^{2}) + 4 \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.10.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x - 5 \cdot 9 x^{2 + 1} - 5 x \cdot 7 + 4 (9 x^{2}) + 4 \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.10.2.3

Add $2$ and $1$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x - 5 \cdot 9 x^{3} - 5 x \cdot 7 + 4 (9 x^{2}) + 4 \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.10.3

Multiply $- 5$ by $9$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x - 45 x^{3} - 5 x \cdot 7 + 4 (9 x^{2}) + 4 \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.10.4

Multiply $7$ by $- 5$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x - 45 x^{3} - 35 x + 4 (9 x^{2}) + 4 \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.10.5

Multiply $9$ by $4$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x - 45 x^{3} - 35 x + 36 x^{2} + 4 \cdot 7}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.10.6

Multiply $4$ by $7$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{35 x^{3} + 35 x - 45 x^{3} - 35 x + 36 x^{2} + 28}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.11

Subtract $45 x^{3}$ from $35 x^{3}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{- 10 x^{3} + 35 x - 35 x + 36 x^{2} + 28}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.12

Subtract $35 x$ from $35 x$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{- 10 x^{3} + 0 + 36 x^{2} + 28}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.13

Add $- 10 x^{3}$ and $0$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{- 10 x^{3} + 36 x^{2} + 28}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.14

Factor $2$ out of $- 10 x^{3} + 36 x^{2} + 28$ .

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Step 4.29.4.6.14.1

Factor $2$ out of $- 10 x^{3}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{2 (- 5 x^{3}) + 36 x^{2} + 28}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.14.2

Factor $2$ out of $36 x^{2}$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{2 (- 5 x^{3}) + 2 (18 x^{2}) + 28}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.14.3

Factor $2$ out of $28$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{2 (- 5 x^{3}) + 2 (18 x^{2}) + 2 (14)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.14.4

Factor $2$ out of $2 (- 5 x^{3}) + 2 (18 x^{2})$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{2 (- 5 x^{3} + 18 x^{2}) + 2 (14)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.6.14.5

Factor $2$ out of $2 (- 5 x^{3} + 18 x^{2}) + 2 (14)$ .

$\frac{x {(x^{2} + 1)}^{\frac{1}{7}} \frac{2 (- 5 x^{3} + 18 x^{2} + 14)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.7

Combine exponents.

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Step 4.29.4.7.1

Combine $\frac{2 (- 5 x^{3} + 18 x^{2} + 14)}{7 {(x^{2} + 1)}^{\frac{6}{7}}}$ and $x$ .

$\frac{\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 {(x^{2} + 1)}^{\frac{6}{7}}} {(x^{2} + 1)}^{\frac{1}{7}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.7.2

Combine $\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 {(x^{2} + 1)}^{\frac{6}{7}}}$ and ${(x^{2} + 1)}^{\frac{1}{7}}$ .

$\frac{\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x {(x^{2} + 1)}^{\frac{1}{7}}}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.8

Reduce the expression $\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x {(x^{2} + 1)}^{\frac{1}{7}}}{7 {(x^{2} + 1)}^{\frac{6}{7}}}$ by cancelling the common factors.

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Step 4.29.4.8.1

Factor ${(x^{2} + 1)}^{\frac{6}{7}}$ out of $2 (- 5 x^{3} + 18 x^{2} + 14) x {(x^{2} + 1)}^{\frac{1}{7}}$ .

$\frac{\frac{{(x^{2} + 1)}^{\frac{6}{7}} (2 (- 5 x^{3} + 18 x^{2} + 14) x {(x^{2} + 1)}^{\frac{- 5}{7}})}{7 {(x^{2} + 1)}^{\frac{6}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.8.2

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

$\frac{\frac{{(x^{2} + 1)}^{\frac{6}{7}} (2 (- 5 x^{3} + 18 x^{2} + 14) x {(x^{2} + 1)}^{\frac{- 5}{7}})}{{(x^{2} + 1)}^{\frac{6}{7}} \cdot 7}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.8.3

Cancel the common factor.

$\frac{\frac{{(x^{2} + 1)}^{\frac{6}{7}} (2 (- 5 x^{3} + 18 x^{2} + 14) x {(x^{2} + 1)}^{\frac{- 5}{7}})}{{(x^{2} + 1)}^{\frac{6}{7}} \cdot 7}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.8.4

Rewrite the expression.

$\frac{\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x {(x^{2} + 1)}^{\frac{- 5}{7}}}{7}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.4.9

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

$\frac{\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 {(x^{2} + 1)}^{- \frac{- 5}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.5

Combine terms.

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Step 4.29.5.1

Move the negative in front of the fraction.

$\frac{\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 {(x^{2} + 1)}^{- - \frac{5}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.5.2

Multiply $- 1$ by $- 1$ .

$\frac{\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 {(x^{2} + 1)}^{1 (\frac{5}{7})}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.5.3

Multiply $\frac{5}{7}$ by $1$ .

$\frac{\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 {(x^{2} + 1)}^{\frac{5}{7}}}}{(5 x - 4) (x^{2} {({(x^{2} + 1)}^{\frac{1}{7}})}^{2})}$

Step 4.29.5.4

Multiply the exponents in ${({(x^{2} + 1)}^{\frac{1}{7}})}^{2}$ .

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Step 4.29.5.4.1

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

$\frac{\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 {(x^{2} + 1)}^{\frac{5}{7}}}}{(5 x - 4) (x^{2} {(x^{2} + 1)}^{\frac{1}{7} \cdot 2})}$

Step 4.29.5.4.2

Combine $\frac{1}{7}$ and $2$ .

$\frac{\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 {(x^{2} + 1)}^{\frac{5}{7}}}}{(5 x - 4) (x^{2} {(x^{2} + 1)}^{\frac{2}{7}})}$

Step 4.29.5.5

Rewrite $\frac{\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 {(x^{2} + 1)}^{\frac{5}{7}}}}{(5 x - 4) x^{2} {(x^{2} + 1)}^{\frac{2}{7}}}$ as a product.

$\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 {(x^{2} + 1)}^{\frac{5}{7}}} \cdot \frac{1}{(5 x - 4) x^{2} {(x^{2} + 1)}^{\frac{2}{7}}}$

Step 4.29.5.6

Multiply $\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 {(x^{2} + 1)}^{\frac{5}{7}}}$ by $\frac{1}{(5 x - 4) x^{2} {(x^{2} + 1)}^{\frac{2}{7}}}$ .

$\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 {(x^{2} + 1)}^{\frac{5}{7}} ((5 x - 4) x^{2} {(x^{2} + 1)}^{\frac{2}{7}})}$

Step 4.29.5.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 ((5 x - 4) x^{2} {(x^{2} + 1)}^{\frac{5}{7} + \frac{2}{7}})}$

Step 4.29.5.8

Combine the numerators over the common denominator.

$\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 ((5 x - 4) x^{2} {(x^{2} + 1)}^{\frac{5 + 2}{7}})}$

Step 4.29.5.9

Add $5$ and $2$ .

$\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 ((5 x - 4) x^{2} {(x^{2} + 1)}^{\frac{7}{7}})}$

Step 4.29.5.10

Cancel the common factor of $7$ .

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Step 4.29.5.10.1

Cancel the common factor.

$\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 ((5 x - 4) x^{2} {(x^{2} + 1)}^{\frac{7}{7}})}$

Step 4.29.5.10.2

Rewrite the expression.

$\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 ((5 x - 4) x^{2} {(x^{2} + 1)}^{1})}$

Step 4.29.5.11

Simplify.

$\frac{2 (- 5 x^{3} + 18 x^{2} + 14) x}{7 ((5 x - 4) x^{2} (x^{2} + 1))}$

Step 4.29.5.12

Cancel the common factor of $x$ and $x^{2}$ .

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Step 4.29.5.12.1

Factor $x$ out of $2 (- 5 x^{3} + 18 x^{2} + 14) x$ .

$\frac{x (2 (- 5 x^{3} + 18 x^{2} + 14))}{7 (5 x - 4) x^{2} (x^{2} + 1)}$

Step 4.29.5.12.2

Cancel the common factors.

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Step 4.29.5.12.2.1

Factor $x$ out of $7 (5 x - 4) x^{2} (x^{2} + 1)$ .

$\frac{x (2 (- 5 x^{3} + 18 x^{2} + 14))}{x (7 (5 x - 4) x (x^{2} + 1))}$

Step 4.29.5.12.2.2

Cancel the common factor.

$\frac{x (2 (- 5 x^{3} + 18 x^{2} + 14))}{x (7 (5 x - 4) x (x^{2} + 1))}$

Step 4.29.5.12.2.3

Rewrite the expression.

$\frac{2 (- 5 x^{3} + 18 x^{2} + 14)}{7 (5 x - 4) x (x^{2} + 1)}$

Step 4.29.6

Reorder terms.

$\frac{2 (- 5 x^{3} + 18 x^{2} + 14)}{7 x (5 x - 4) (x^{2} + 1)}$

Step 4.29.7

Factor $- 1$ out of $- 5 x^{3}$ .

$\frac{2 (- (5 x^{3}) + 18 x^{2} + 14)}{7 x (5 x - 4) (x^{2} + 1)}$

Step 4.29.8

Factor $- 1$ out of $18 x^{2}$ .

$\frac{2 (- (5 x^{3}) - (- 18 x^{2}) + 14)}{7 x (5 x - 4) (x^{2} + 1)}$

Step 4.29.9

Factor $- 1$ out of $- (5 x^{3}) - (- 18 x^{2})$ .

$\frac{2 (- (5 x^{3} - 18 x^{2}) + 14)}{7 x (5 x - 4) (x^{2} + 1)}$

Step 4.29.10

Rewrite $14$ as $- 1 (- 14)$ .

$\frac{2 (- (5 x^{3} - 18 x^{2}) - 1 (- 14))}{7 x (5 x - 4) (x^{2} + 1)}$

Step 4.29.11

Factor $- 1$ out of $- (5 x^{3} - 18 x^{2}) - 1 (- 14)$ .

$\frac{2 (- (5 x^{3} - 18 x^{2} - 14))}{7 x (5 x - 4) (x^{2} + 1)}$

Step 4.29.12

Rewrite $- (5 x^{3} - 18 x^{2} - 14)$ as $- 1 (5 x^{3} - 18 x^{2} - 14)$ .

$\frac{2 (- 1 (5 x^{3} - 18 x^{2} - 14))}{7 x (5 x - 4) (x^{2} + 1)}$

Step 4.29.13

Move the negative in front of the fraction.

$- \frac{2 (5 x^{3} - 18 x^{2} - 14)}{7 x (5 x - 4) (x^{2} + 1)}$

Step 5

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

$y' = - \frac{2 (5 x^{3} - 18 x^{2} - 14)}{7 x (5 x - 4) (x^{2} + 1)}$

Step 6

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

$\frac{d y}{d x} = - \frac{2 (5 x^{3} - 18 x^{2} - 14)}{7 x (5 x - 4) (x^{2} + 1)}$