Calculus Examples

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

Step 1

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 $f (x) = {(x^{2} + 1)}^{\frac{3}{2}}$ and $g (x) = x$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = x^{2} + 1$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $3$ .

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

Step 7

Combine fractions.

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Step 7.1

Combine $\frac{3}{2}$ and ${(x^{2} + 1)}^{\frac{1}{2}}$ .

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

Step 7.2

Combine $\frac{3 {(x^{2} + 1)}^{\frac{1}{2}}}{2}$ and $x$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

$\frac{\frac{3 {(x^{2} + 1)}^{\frac{1}{2}} x}{2} (2 x + 0) - {(x^{2} + 1)}^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 11

Combine fractions.

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Step 11.1

Add $2 x$ and $0$ .

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

Step 11.2

Combine $2$ and $\frac{3 {(x^{2} + 1)}^{\frac{1}{2}} x}{2}$ .

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

Step 11.3

Multiply $3$ by $2$ .

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

Step 11.4

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Add $1$ and $1$ .

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

Step 16

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

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

Step 17

Cancel the common factors.

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Step 17.1

Factor $2$ out of $2$ .

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

Step 17.2

Cancel the common factor.

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

Step 17.3

Rewrite the expression.

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

Step 17.4

Divide $3 {(x^{2} + 1)}^{\frac{1}{2}} x^{2}$ by $1$ .

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

Step 18

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

$\frac{3 {(x^{2} + 1)}^{\frac{1}{2}} x^{2} - {(x^{2} + 1)}^{\frac{3}{2}} \cdot 1}{x^{2}}$

Step 19

Multiply $- 1$ by $1$ .

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

Step 20

Simplify the numerator.

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Step 20.1

Factor ${(x^{2} + 1)}^{\frac{1}{2}}$ out of $3 {(x^{2} + 1)}^{\frac{1}{2}} x^{2} - {(x^{2} + 1)}^{\frac{3}{2}}$ .

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Step 20.1.1

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

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

Step 20.1.2

Factor ${(x^{2} + 1)}^{\frac{1}{2}}$ out of $3 x^{2} {(x^{2} + 1)}^{\frac{1}{2}}$ .

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

Step 20.1.3

Factor ${(x^{2} + 1)}^{\frac{1}{2}}$ out of $- {(x^{2} + 1)}^{\frac{3}{2}}$ .

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

Step 20.1.4

Factor ${(x^{2} + 1)}^{\frac{1}{2}}$ out of ${(x^{2} + 1)}^{\frac{1}{2}} (3 x^{2}) + {(x^{2} + 1)}^{\frac{1}{2}} (- {(x^{2} + 1)}^{\frac{2}{2}})$ .

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

Step 20.2

Divide $2$ by $2$ .

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

Step 20.3

Simplify.

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

Step 20.4

Apply the distributive property.

$\frac{{(x^{2} + 1)}^{\frac{1}{2}} (3 x^{2} - x^{2} - 1 \cdot 1)}{x^{2}}$

Step 20.5

Multiply $- 1$ by $1$ .

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

Step 20.6

Subtract $x^{2}$ from $3 x^{2}$ .

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