Calculus Examples

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

Step 1

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 $g (r) = \sqrt{r^{2} + 1}$ .

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

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify terms.

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Step 12.1

Add $2 r$ and $0$ .

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

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

Cancel the common factor.

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

Step 12.5

Rewrite the expression.

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

Step 13

Simplify.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Simplify each term.

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Step 13.1.1.1

Multiply $\sqrt{r^{2} + 1}$ by $1$ .

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

Step 13.1.1.2

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

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Step 13.1.1.2.1

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

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

Step 13.1.1.2.2

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

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

Step 13.1.1.2.3

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

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

Step 13.1.1.2.4

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

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

Step 13.1.1.2.5

Add $1$ and $1$ .

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

Step 13.1.2

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

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

Step 13.1.3

Combine the numerators over the common denominator.

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

Step 13.1.4

Simplify the numerator.

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Step 13.1.4.1

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

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Step 13.1.4.1.1

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

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

Step 13.1.4.1.2

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

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

Step 13.1.4.1.3

Combine the numerators over the common denominator.

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

Step 13.1.4.1.4

Add $1$ and $1$ .

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

Step 13.1.4.1.5

Cancel the common factor of $2$ .

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Step 13.1.4.1.5.1

Cancel the common factor.

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

Step 13.1.4.1.5.2

Rewrite the expression.

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

Step 13.1.4.2

Simplify.

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

Step 13.1.4.3

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

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

Step 13.1.4.4

Add $0$ and $1$ .

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

Step 13.2

Combine terms.

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Step 13.2.1

Rewrite ${\sqrt{r^{2} + 1}}^{2}$ as $r^{2} + 1$ .

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Step 13.2.1.1

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

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

Step 13.2.1.2

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

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

Step 13.2.1.3

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

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

Step 13.2.1.4

Cancel the common factor of $2$ .

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Step 13.2.1.4.1

Cancel the common factor.

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

Step 13.2.1.4.2

Rewrite the expression.

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

Step 13.2.1.5

Simplify.

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

Step 13.2.2

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 13.2.3

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 13.2.4

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

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Step 13.2.4.1

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

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Step 13.2.4.1.1

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

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

Step 13.2.4.1.2

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

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

Step 13.2.4.2

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

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

Step 13.2.4.3

Combine the numerators over the common denominator.

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

Step 13.2.4.4

Add $1$ and $2$ .

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