Calculus Examples

$x \sqrt{1 + x^{2}}$

Step 1

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

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

Step 2

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

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

Tap for more steps...

Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 9

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

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

Step 10

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

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

Step 11

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

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

Step 12

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

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

Step 13

Combine fractions.

Tap for more steps...

Step 13.1

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

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

Step 13.2

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

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

Step 14

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

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

Step 15

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

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

Step 16

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

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

Step 17

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 17.1

Add $1$ and $1$ .

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

Step 17.2

Cancel the common factor.

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

Step 17.3

Rewrite the expression.

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

Step 18

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

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

Step 19

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

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

Step 20

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

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

Step 21

Combine the numerators over the common denominator.

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

Step 22

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

Tap for more steps...

Step 22.1

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

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

Step 22.2

Combine the numerators over the common denominator.

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

Step 22.3

Add $1$ and $1$ .

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

Step 22.4

Divide $2$ by $2$ .

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

Step 23

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

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

Step 24

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

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

Step 25

Reorder terms.

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