Calculus Examples

$\ln (\frac{{(2 y + 1)}^{5}}{\sqrt{y^{2} + 1}})$

Step 1

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

$\frac{d}{d y} [\ln (\frac{{(2 y + 1)}^{5}}{{(y^{2} + 1)}^{\frac{1}{2}}})]$

Step 2

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

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Step 2.1

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

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d y} [\frac{{(2 y + 1)}^{5}}{{(y^{2} + 1)}^{\frac{1}{2}}}]$

Step 2.2

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

$\frac{1}{u_{1}} \frac{d}{d y} [\frac{{(2 y + 1)}^{5}}{{(y^{2} + 1)}^{\frac{1}{2}}}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $\frac{{(2 y + 1)}^{5}}{{(y^{2} + 1)}^{\frac{1}{2}}}$ .

$\frac{1}{\frac{{(2 y + 1)}^{5}}{{(y^{2} + 1)}^{\frac{1}{2}}}} \frac{d}{d y} [\frac{{(2 y + 1)}^{5}}{{(y^{2} + 1)}^{\frac{1}{2}}}]$

Step 3

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

$1 \frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \frac{d}{d y} [\frac{{(2 y + 1)}^{5}}{{(y^{2} + 1)}^{\frac{1}{2}}}]$

Step 4

Multiply $\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}}$ by $1$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \frac{d}{d y} [\frac{{(2 y + 1)}^{5}}{{(y^{2} + 1)}^{\frac{1}{2}}}]$

Step 5

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

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

Step 6

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

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Step 6.1

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

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Cancel the common factor.

Step 6.2.2

Rewrite the expression.

Step 7

Simplify.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 5$ .

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

Step 8.3

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

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

Step 9

Differentiate.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

Multiply $2$ by $1$ .

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

Step 9.6

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

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

Step 9.7

Simplify the expression.

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Step 9.7.1

Add $2$ and $0$ .

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

Step 9.7.2

Multiply $2$ by $5$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} \frac{d}{d y} [{(y^{2} + 1)}^{\frac{1}{2}}]}{y^{2} + 1}$

Step 10

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

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Step 10.1

To apply the Chain Rule, set $u_{3}$ as $y^{2} + 1$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} (\frac{d}{d u_{3}} [{u_{3}}^{\frac{1}{2}}] \frac{d}{d y} [y^{2} + 1])}{y^{2} + 1}$

Step 10.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} (\frac{1}{2} {u_{3}}^{\frac{1}{2} - 1} \frac{d}{d y} [y^{2} + 1])}{y^{2} + 1}$

Step 10.3

Replace all occurrences of $u_{3}$ with $y^{2} + 1$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} (\frac{1}{2} {(y^{2} + 1)}^{\frac{1}{2} - 1} \frac{d}{d y} [y^{2} + 1])}{y^{2} + 1}$

Step 11

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} (\frac{1}{2} {(y^{2} + 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d y} [y^{2} + 1])}{y^{2} + 1}$

Step 12

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

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} (\frac{1}{2} {(y^{2} + 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d y} [y^{2} + 1])}{y^{2} + 1}$

Step 13

Combine the numerators over the common denominator.

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} (\frac{1}{2} {(y^{2} + 1)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d y} [y^{2} + 1])}{y^{2} + 1}$

Step 14

Simplify the numerator.

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Step 14.1

Multiply $- 1$ by $2$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} (\frac{1}{2} {(y^{2} + 1)}^{\frac{1 - 2}{2}} \frac{d}{d y} [y^{2} + 1])}{y^{2} + 1}$

Step 14.2

Subtract $2$ from $1$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} (\frac{1}{2} {(y^{2} + 1)}^{\frac{- 1}{2}} \frac{d}{d y} [y^{2} + 1])}{y^{2} + 1}$

Step 15

Combine fractions.

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Step 15.1

Move the negative in front of the fraction.

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} (\frac{1}{2} {(y^{2} + 1)}^{- \frac{1}{2}} \frac{d}{d y} [y^{2} + 1])}{y^{2} + 1}$

Step 15.2

Combine $\frac{1}{2}$ and ${(y^{2} + 1)}^{- \frac{1}{2}}$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} (\frac{{(y^{2} + 1)}^{- \frac{1}{2}}}{2} \frac{d}{d y} [y^{2} + 1])}{y^{2} + 1}$

Step 15.3

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

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} (\frac{1}{2 {(y^{2} + 1)}^{\frac{1}{2}}} \frac{d}{d y} [y^{2} + 1])}{y^{2} + 1}$

Step 15.4

Combine $\frac{1}{2 {(y^{2} + 1)}^{\frac{1}{2}}}$ and ${(2 y + 1)}^{5}$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - \frac{{(2 y + 1)}^{5}}{2 {(y^{2} + 1)}^{\frac{1}{2}}} \frac{d}{d y} [y^{2} + 1]}{y^{2} + 1}$

Step 16

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

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - \frac{{(2 y + 1)}^{5}}{2 {(y^{2} + 1)}^{\frac{1}{2}}} (\frac{d}{d y} [y^{2}] + \frac{d}{d y} [1])}{y^{2} + 1}$

Step 17

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

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - \frac{{(2 y + 1)}^{5}}{2 {(y^{2} + 1)}^{\frac{1}{2}}} (2 y + \frac{d}{d y} [1])}{y^{2} + 1}$

Step 18

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

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - \frac{{(2 y + 1)}^{5}}{2 {(y^{2} + 1)}^{\frac{1}{2}}} (2 y + 0)}{y^{2} + 1}$

Step 19

Simplify terms.

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Step 19.1

Add $2 y$ and $0$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - \frac{{(2 y + 1)}^{5}}{2 {(y^{2} + 1)}^{\frac{1}{2}}} (2 y)}{y^{2} + 1}$

Step 19.2

Multiply $2$ by $- 1$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - 2 \frac{{(2 y + 1)}^{5}}{2 {(y^{2} + 1)}^{\frac{1}{2}}} y}{y^{2} + 1}$

Step 19.3

Combine $- 2$ and $\frac{{(2 y + 1)}^{5}}{2 {(y^{2} + 1)}^{\frac{1}{2}}}$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} + \frac{- 2 {(2 y + 1)}^{5}}{2 {(y^{2} + 1)}^{\frac{1}{2}}} y}{y^{2} + 1}$

Step 19.4

Combine $\frac{- 2 {(2 y + 1)}^{5}}{2 {(y^{2} + 1)}^{\frac{1}{2}}}$ and $y$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} + \frac{- 2 {(2 y + 1)}^{5} y}{2 {(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 19.5

Factor $2$ out of $- 2 {(2 y + 1)}^{5} y$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} + \frac{2 (- {(2 y + 1)}^{5} y)}{2 {(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 20

Cancel the common factors.

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Step 20.1

Factor $2$ out of $2 {(y^{2} + 1)}^{\frac{1}{2}}$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} + \frac{2 (- {(2 y + 1)}^{5} y)}{2 ({(y^{2} + 1)}^{\frac{1}{2}})}}{y^{2} + 1}$

Step 20.2

Cancel the common factor.

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} + \frac{2 (- {(2 y + 1)}^{5} y)}{2 {(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 20.3

Rewrite the expression.

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} + \frac{- {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 21

Move the negative in front of the fraction.

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} - \frac{{(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 22

To write $10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4}$ as a fraction with a common denominator, multiply by $\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(y^{2} + 1)}^{\frac{1}{2}}}$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{\frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} {(y^{2} + 1)}^{\frac{1}{2}}}{{(y^{2} + 1)}^{\frac{1}{2}}} - \frac{{(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 23

Combine the numerators over the common denominator.

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{\frac{10 {(y^{2} + 1)}^{\frac{1}{2}} {(2 y + 1)}^{4} {(y^{2} + 1)}^{\frac{1}{2}} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 24

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

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Step 24.1

Move ${(y^{2} + 1)}^{\frac{1}{2}}$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{\frac{10 ({(y^{2} + 1)}^{\frac{1}{2}} {(y^{2} + 1)}^{\frac{1}{2}}) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 24.2

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

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{\frac{10 {(y^{2} + 1)}^{\frac{1}{2} + \frac{1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 24.3

Combine the numerators over the common denominator.

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{\frac{10 {(y^{2} + 1)}^{\frac{1 + 1}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 24.4

Add $1$ and $1$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{\frac{10 {(y^{2} + 1)}^{\frac{2}{2}} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 24.5

Divide $2$ by $2$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{\frac{10 {(y^{2} + 1)}^{1} {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 25

Simplify $10 {(y^{2} + 1)}^{1} {(2 y + 1)}^{4}$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{\frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$

Step 26

Rewrite $\frac{\frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}}}{y^{2} + 1}$ as a product.

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} (\frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}} \cdot \frac{1}{y^{2} + 1})$

Step 27

Multiply $\frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}}}$ by $\frac{1}{y^{2} + 1}$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}} (y^{2} + 1)}$

Step 28

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

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Step 28.1

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

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Step 28.1.1

Raise $y^{2} + 1$ to the power of $1$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2}} {(y^{2} + 1)}^{1}}$

Step 28.1.2

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

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2} + 1}}$

Step 28.2

Write $1$ as a fraction with a common denominator.

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 28.3

Combine the numerators over the common denominator.

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{1 + 2}{2}}}$

Step 28.4

Add $1$ and $2$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}} \cdot \frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{3}{2}}}$

Step 29

Multiply $\frac{{(y^{2} + 1)}^{\frac{1}{2}}}{{(2 y + 1)}^{5}}$ by $\frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(y^{2} + 1)}^{\frac{3}{2}}}$ .

$\frac{{(y^{2} + 1)}^{\frac{1}{2}} (10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y)}{{(2 y + 1)}^{5} {(y^{2} + 1)}^{\frac{3}{2}}}$

Step 30

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

$\frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(2 y + 1)}^{5} {(y^{2} + 1)}^{\frac{3}{2}} {(y^{2} + 1)}^{- \frac{1}{2}}}$

Step 31

Simplify the denominator.

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Step 31.1

Multiply ${(y^{2} + 1)}^{\frac{3}{2}}$ by ${(y^{2} + 1)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 31.1.1

Move ${(y^{2} + 1)}^{- \frac{1}{2}}$ .

$\frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(2 y + 1)}^{5} ({(y^{2} + 1)}^{- \frac{1}{2}} {(y^{2} + 1)}^{\frac{3}{2}})}$

Step 31.1.2

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

$\frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(2 y + 1)}^{5} {(y^{2} + 1)}^{- \frac{1}{2} + \frac{3}{2}}}$

Step 31.1.3

Combine the numerators over the common denominator.

$\frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(2 y + 1)}^{5} {(y^{2} + 1)}^{\frac{- 1 + 3}{2}}}$

Step 31.1.4

Add $- 1$ and $3$ .

$\frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(2 y + 1)}^{5} {(y^{2} + 1)}^{\frac{2}{2}}}$

Step 31.1.5

Divide $2$ by $2$ .

$\frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(2 y + 1)}^{5} {(y^{2} + 1)}^{1}}$

Step 31.2

Simplify ${(2 y + 1)}^{5} {(y^{2} + 1)}^{1}$ .

$\frac{10 (y^{2} + 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32

Simplify.

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Step 32.1

Apply the distributive property.

$\frac{(10 y^{2} + 10 \cdot 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2

Simplify the numerator.

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Step 32.2.1

Factor ${(2 y + 1)}^{4}$ out of $(10 y^{2} + 10 \cdot 1) {(2 y + 1)}^{4} - {(2 y + 1)}^{5} y$ .

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Step 32.2.1.1

Factor ${(2 y + 1)}^{4}$ out of $(10 y^{2} + 10 \cdot 1) {(2 y + 1)}^{4}$ .

$\frac{{(2 y + 1)}^{4} (10 y^{2} + 10 \cdot 1) - {(2 y + 1)}^{5} y}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2.1.2

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

$\frac{{(2 y + 1)}^{4} (10 y^{2} + 10 \cdot 1) + {(2 y + 1)}^{4} (- (2 y + 1) y)}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2.1.3

Factor ${(2 y + 1)}^{4}$ out of ${(2 y + 1)}^{4} (10 y^{2} + 10 \cdot 1) + {(2 y + 1)}^{4} (- (2 y + 1) y)$ .

$\frac{{(2 y + 1)}^{4} (10 y^{2} + 10 \cdot 1 - (2 y + 1) y)}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2.2

Multiply $10$ by $1$ .

$\frac{{(2 y + 1)}^{4} (10 y^{2} + 10 - (2 y + 1) y)}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2.3

Apply the distributive property.

$\frac{{(2 y + 1)}^{4} (10 y^{2} + 10 + (- (2 y) - 1 \cdot 1) y)}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2.4

Multiply $2$ by $- 1$ .

$\frac{{(2 y + 1)}^{4} (10 y^{2} + 10 + (- 2 y - 1 \cdot 1) y)}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2.5

Multiply $- 1$ by $1$ .

$\frac{{(2 y + 1)}^{4} (10 y^{2} + 10 + (- 2 y - 1) y)}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2.6

Apply the distributive property.

$\frac{{(2 y + 1)}^{4} (10 y^{2} + 10 - 2 y \cdot y - 1 y)}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2.7

Multiply $y$ by $y$ by adding the exponents.

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Step 32.2.7.1

Move $y$ .

$\frac{{(2 y + 1)}^{4} (10 y^{2} + 10 - 2 (y \cdot y) - 1 y)}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2.7.2

Multiply $y$ by $y$ .

$\frac{{(2 y + 1)}^{4} (10 y^{2} + 10 - 2 y^{2} - 1 y)}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2.8

Rewrite $- 1 y$ as $- y$ .

$\frac{{(2 y + 1)}^{4} (10 y^{2} + 10 - 2 y^{2} - y)}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2.9

Subtract $2 y^{2}$ from $10 y^{2}$ .

$\frac{{(2 y + 1)}^{4} (8 y^{2} + 10 - y)}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.2.10

Reorder terms.

$\frac{{(2 y + 1)}^{4} (8 y^{2} - y + 10)}{{(2 y + 1)}^{5} (y^{2} + 1)}$

Step 32.3

Cancel the common factors.

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Step 32.3.1

Factor ${(2 y + 1)}^{4}$ out of ${(2 y + 1)}^{5} (y^{2} + 1)$ .

$\frac{{(2 y + 1)}^{4} (8 y^{2} - y + 10)}{{(2 y + 1)}^{4} ((2 y + 1) (y^{2} + 1))}$

Step 32.3.2

Cancel the common factor.

$\frac{{(2 y + 1)}^{4} (8 y^{2} - y + 10)}{{(2 y + 1)}^{4} ((2 y + 1) (y^{2} + 1))}$

Step 32.3.3

Rewrite the expression.

$\frac{8 y^{2} - y + 10}{(2 y + 1) (y^{2} + 1)}$