Calculus Examples

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

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Evaluate $\frac{d}{d x} [\sqrt{1 + x^{2}}]$ .

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 1$ by $2$ .

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

Step 2.9.2

Subtract $2$ from $1$ .

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

Step 2.10

Move the negative in front of the fraction.

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

Step 2.11

Add $0$ and $2 x$ .

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

Cancel the common factor.

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

Step 2.17

Rewrite the expression.

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

Step 3

Reorder terms.

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