Calculus Examples

$y = \frac{r}{\sqrt{r^{2} + 1}}$

Step 1

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

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

Step 2

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} \frac{d}{d r} [r] - r \frac{d}{d r} [{(r^{2} + 1)}^{\frac{1}{2}}]}{{({(r^{2} + 1)}^{\frac{1}{2}})}^{2}}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} \frac{d}{d r} [r] - r \frac{d}{d r} [{(r^{2} + 1)}^{\frac{1}{2}}]}{{(r^{2} + 1)}^{1}}$

Step 4

Simplify.

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} \frac{d}{d r} [r] - r \frac{d}{d r} [{(r^{2} + 1)}^{\frac{1}{2}}]}{r^{2} + 1}$

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

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

Step 5.2

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - r \frac{d}{d r} [{(r^{2} + 1)}^{\frac{1}{2}}]}{r^{2} + 1}$

Step 6

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

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Step 6.1

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - r (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d r} [r^{2} + 1])}{r^{2} + 1}$

Step 6.2

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - r (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d r} [r^{2} + 1])}{r^{2} + 1}$

Step 6.3

Replace all occurrences of $u$ with $r^{2} + 1$ .

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - r (\frac{1}{2} {(r^{2} + 1)}^{\frac{1}{2} - 1} \frac{d}{d r} [r^{2} + 1])}{r^{2} + 1}$

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - r (\frac{1}{2} {(r^{2} + 1)}^{\frac{1 - 2}{2}} \frac{d}{d r} [r^{2} + 1])}{r^{2} + 1}$

Step 10.2

Subtract $2$ from $1$ .

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - r (\frac{1}{2} {(r^{2} + 1)}^{\frac{- 1}{2}} \frac{d}{d r} [r^{2} + 1])}{r^{2} + 1}$

Step 11

Combine fractions.

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Step 11.1

Move the negative in front of the fraction.

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - r (\frac{1}{2} {(r^{2} + 1)}^{- \frac{1}{2}} \frac{d}{d r} [r^{2} + 1])}{r^{2} + 1}$

Step 11.2

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - r (\frac{{(r^{2} + 1)}^{- \frac{1}{2}}}{2} \frac{d}{d r} [r^{2} + 1])}{r^{2} + 1}$

Step 11.3

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - r (\frac{1}{2 {(r^{2} + 1)}^{\frac{1}{2}}} \frac{d}{d r} [r^{2} + 1])}{r^{2} + 1}$

Step 11.4

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - \frac{r}{2 {(r^{2} + 1)}^{\frac{1}{2}}} \frac{d}{d r} [r^{2} + 1]}{r^{2} + 1}$

Step 12

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - \frac{r}{2 {(r^{2} + 1)}^{\frac{1}{2}}} (\frac{d}{d r} [r^{2}] + \frac{d}{d r} [1])}{r^{2} + 1}$

Step 13

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - \frac{r}{2 {(r^{2} + 1)}^{\frac{1}{2}}} (2 r + \frac{d}{d r} [1])}{r^{2} + 1}$

Step 14

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - \frac{r}{2 {(r^{2} + 1)}^{\frac{1}{2}}} (2 r + 0)}{r^{2} + 1}$

Step 15

Combine fractions.

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Step 15.1

Add $2 r$ and $0$ .

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - \frac{r}{2 {(r^{2} + 1)}^{\frac{1}{2}}} (2 r)}{r^{2} + 1}$

Step 15.2

Multiply $2$ by $- 1$ .

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - 2 \frac{r}{2 {(r^{2} + 1)}^{\frac{1}{2}}} r}{r^{2} + 1}$

Step 15.3

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} + \frac{- 2 r}{2 {(r^{2} + 1)}^{\frac{1}{2}}} r}{r^{2} + 1}$

Step 15.4

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} + \frac{- 2 r \cdot r}{2 {(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 16

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} + \frac{- 2 (r^{1} r)}{2 {(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 17

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} + \frac{- 2 (r^{1} r^{1})}{2 {(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 18

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} + \frac{- 2 r^{1 + 1}}{2 {(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 19

Add $1$ and $1$ .

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} + \frac{- 2 r^{2}}{2 {(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 20

Factor $2$ out of $- 2 r^{2}$ .

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} + \frac{2 (- r^{2})}{2 {(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 21

Cancel the common factors.

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Step 21.1

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

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} + \frac{2 (- r^{2})}{2 ({(r^{2} + 1)}^{\frac{1}{2}})}}{r^{2} + 1}$

Step 21.2

Cancel the common factor.

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} + \frac{2 (- r^{2})}{2 {(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 21.3

Rewrite the expression.

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} + \frac{- r^{2}}{{(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 22

Move the negative in front of the fraction.

$\frac{{(r^{2} + 1)}^{\frac{1}{2}} - \frac{r^{2}}{{(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 23

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

$\frac{\frac{{(r^{2} + 1)}^{\frac{1}{2}} {(r^{2} + 1)}^{\frac{1}{2}}}{{(r^{2} + 1)}^{\frac{1}{2}}} - \frac{r^{2}}{{(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 24

Combine the numerators over the common denominator.

$\frac{\frac{{(r^{2} + 1)}^{\frac{1}{2}} {(r^{2} + 1)}^{\frac{1}{2}} - r^{2}}{{(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 25

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

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Step 25.1

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

$\frac{\frac{{(r^{2} + 1)}^{\frac{1}{2} + \frac{1}{2}} - r^{2}}{{(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 25.2

Combine the numerators over the common denominator.

$\frac{\frac{{(r^{2} + 1)}^{\frac{1 + 1}{2}} - r^{2}}{{(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 25.3

Add $1$ and $1$ .

$\frac{\frac{{(r^{2} + 1)}^{\frac{2}{2}} - r^{2}}{{(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 25.4

Divide $2$ by $2$ .

$\frac{\frac{{(r^{2} + 1)}^{1} - r^{2}}{{(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 26

Simplify ${(r^{2} + 1)}^{1}$ .

$\frac{\frac{r^{2} + 1 - r^{2}}{{(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 27

Subtract $r^{2}$ from $r^{2}$ .

$\frac{\frac{0 + 1}{{(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 28

Add $0$ and $1$ .

$\frac{\frac{1}{{(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$

Step 29

Rewrite $\frac{\frac{1}{{(r^{2} + 1)}^{\frac{1}{2}}}}{r^{2} + 1}$ as a product.

$\frac{1}{{(r^{2} + 1)}^{\frac{1}{2}}} \cdot \frac{1}{r^{2} + 1}$

Step 30

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

$\frac{1}{{(r^{2} + 1)}^{\frac{1}{2}} (r^{2} + 1)}$

Step 31

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

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Step 31.1

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

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Step 31.1.1

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

$\frac{1}{{(r^{2} + 1)}^{\frac{1}{2}} {(r^{2} + 1)}^{1}}$

Step 31.1.2

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

$\frac{1}{{(r^{2} + 1)}^{\frac{1}{2} + 1}}$

Step 31.2

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

$\frac{1}{{(r^{2} + 1)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 31.3

Combine the numerators over the common denominator.

$\frac{1}{{(r^{2} + 1)}^{\frac{1 + 2}{2}}}$

Step 31.4

Add $1$ and $2$ .

$\frac{1}{{(r^{2} + 1)}^{\frac{3}{2}}}$