Calculus Examples

$y = (2 x + 3) \sqrt{x^{2} + 3 x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 3 x}$ as ${(x^{2} + 3 x)}^{\frac{1}{2}}$ .

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

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

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

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 $g (x) = x^{2} + 3 x$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x^{2} + 3 x$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $x^{2} + 3 x$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

Move ${(x^{2} + 3 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9

By the Sum Rule, the derivative of $x^{2} + 3 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x]$ .

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

Step 10

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

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

Step 11

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 12

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

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

Step 13

Multiply $3$ by $1$ .

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

Step 14

Raise $2 x + 3$ to the power of $1$ .

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

Step 15

Raise $2 x + 3$ to the power of $1$ .

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

Step 16

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

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

Step 17

Combine fractions.

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Step 17.1

Add $1$ and $1$ .

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

Step 17.2

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

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

Step 18

By the Sum Rule, the derivative of $2 x + 3$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [3]$ .

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

Step 19

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

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

Step 20

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

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

Step 21

Multiply $2$ by $1$ .

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

Step 22

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{{(2 x + 3)}^{2}}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}} + {(x^{2} + 3 x)}^{\frac{1}{2}} (2 + 0)$

Step 23

Simplify the expression.

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Step 23.1

Add $2$ and $0$ .

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

Step 23.2

Move $2$ to the left of ${(x^{2} + 3 x)}^{\frac{1}{2}}$ .

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

Step 24

To write $2 {(x^{2} + 3 x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$ .

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

Step 25

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

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

Step 26

Combine the numerators over the common denominator.

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

Step 27

Multiply $2$ by $2$ .

$\frac{{(2 x + 3)}^{2} + 4 {(x^{2} + 3 x)}^{\frac{1}{2}} {(x^{2} + 3 x)}^{\frac{1}{2}}}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 28

Multiply ${(x^{2} + 3 x)}^{\frac{1}{2}}$ by ${(x^{2} + 3 x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 28.1

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

$\frac{{(2 x + 3)}^{2} + 4 ({(x^{2} + 3 x)}^{\frac{1}{2}} {(x^{2} + 3 x)}^{\frac{1}{2}})}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 28.2

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

$\frac{{(2 x + 3)}^{2} + 4 {(x^{2} + 3 x)}^{\frac{1}{2} + \frac{1}{2}}}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 28.3

Combine the numerators over the common denominator.

$\frac{{(2 x + 3)}^{2} + 4 {(x^{2} + 3 x)}^{\frac{1 + 1}{2}}}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 28.4

Add $1$ and $1$ .

$\frac{{(2 x + 3)}^{2} + 4 {(x^{2} + 3 x)}^{\frac{2}{2}}}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 28.5

Divide $2$ by $2$ .

$\frac{{(2 x + 3)}^{2} + 4 {(x^{2} + 3 x)}^{1}}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 29

Simplify $4 {(x^{2} + 3 x)}^{1}$ .

$\frac{{(2 x + 3)}^{2} + 4 (x^{2} + 3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30

Simplify.

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Step 30.1

Apply the distributive property.

$\frac{{(2 x + 3)}^{2} + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2

Simplify the numerator.

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Step 30.2.1

Simplify each term.

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Step 30.2.1.1

Rewrite ${(2 x + 3)}^{2}$ as $(2 x + 3) (2 x + 3)$ .

$\frac{(2 x + 3) (2 x + 3) + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.1.2

Expand $(2 x + 3) (2 x + 3)$ using the FOIL Method.

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Step 30.2.1.2.1

Apply the distributive property.

$\frac{2 x (2 x + 3) + 3 (2 x + 3) + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.1.2.2

Apply the distributive property.

$\frac{2 x (2 x) + 2 x \cdot 3 + 3 (2 x + 3) + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.1.2.3

Apply the distributive property.

$\frac{2 x (2 x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.1.3

Simplify and combine like terms.

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Step 30.2.1.3.1

Simplify each term.

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Step 30.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 \cdot 2 x \cdot x + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 30.2.1.3.1.2.1

Move $x$ .

$\frac{2 \cdot 2 (x \cdot x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.1.3.1.2.2

Multiply $x$ by $x$ .

$\frac{2 \cdot 2 x^{2} + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.1.3.1.3

Multiply $2$ by $2$ .

$\frac{4 x^{2} + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.1.3.1.4

Multiply $3$ by $2$ .

$\frac{4 x^{2} + 6 x + 3 (2 x) + 3 \cdot 3 + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.1.3.1.5

Multiply $2$ by $3$ .

$\frac{4 x^{2} + 6 x + 6 x + 3 \cdot 3 + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.1.3.1.6

Multiply $3$ by $3$ .

$\frac{4 x^{2} + 6 x + 6 x + 9 + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.1.3.2

Add $6 x$ and $6 x$ .

$\frac{4 x^{2} + 12 x + 9 + 4 x^{2} + 4 (3 x)}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.1.4

Multiply $3$ by $4$ .

$\frac{4 x^{2} + 12 x + 9 + 4 x^{2} + 12 x}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.2

Add $4 x^{2}$ and $4 x^{2}$ .

$\frac{8 x^{2} + 12 x + 9 + 12 x}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$

Step 30.2.3

Add $12 x$ and $12 x$ .

$\frac{8 x^{2} + 24 x + 9}{2 {(x^{2} + 3 x)}^{\frac{1}{2}}}$