Calculus Examples

$2 x \sqrt{x + 1}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

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

Step 1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x {(x + 1)}^{\frac{1}{2}}$ with respect to $x$ is $2 \frac{d}{d x} [x {(x + 1)}^{\frac{1}{2}}]$ .

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

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

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

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 $g (x) = x + 1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify the expression.

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Step 12.1

Add $1$ and $0$ .

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

Step 12.2

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

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

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

Step 17

Combine the numerators over the common denominator.

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

Step 18

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

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Step 18.1

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

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

Step 18.2

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

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

Step 18.3

Combine the numerators over the common denominator.

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

Step 18.4

Add $1$ and $1$ .

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

Step 18.5

Divide $2$ by $2$ .

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

Step 19

Simplify the expression.

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Step 19.1

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

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

Step 19.2

Move $2$ to the left of $x + 1$ .

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

Step 20

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

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

Step 21

Cancel the common factor.

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

Step 22

Rewrite the expression.

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

Step 23

Simplify.

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Step 23.1

Apply the distributive property.

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

Step 23.2

Simplify the numerator.

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Step 23.2.1

Multiply $2$ by $1$ .

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

Step 23.2.2

Add $x$ and $2 x$ .

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