Calculus Examples

$x + \sqrt{x + 1}$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

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

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

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 $g (x) = x + 1$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x + 1$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $x + 1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 1$ by $2$ .

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

Step 2.9.2

Subtract $2$ from $1$ .

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

Step 2.10

Move the negative in front of the fraction.

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

Step 2.11

Add $1$ and $0$ .

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

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

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