Calculus Examples

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

Step 1

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 (x + 2)}{2 {(x + 3)}^{\frac{1}{2}}}$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [\frac{x + 2}{{(x + 3)}^{\frac{1}{2}}}]$ .

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x + 2$ and $g (x) = {(x + 3)}^{\frac{1}{2}}$ .

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

Step 3

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

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Step 3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify.

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{3}{2} \cdot \frac{{(x + 3)}^{\frac{1}{2}} (1 + 0) - (x + 2) \frac{d}{d x} [{(x + 3)}^{\frac{1}{2}}]}{x + 3}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Add $1$ and $0$ .

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

Step 5.4.2

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

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

Step 6

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 $g (x) = x + 3$ .

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Step 6.1

To apply the Chain Rule, set $u_{1}$ as $x + 3$ .

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

Step 6.2

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

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

Step 6.3

Replace all occurrences of $u_{1}$ with $x + 3$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

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

Step 10.2

Subtract $2$ from $1$ .

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

Step 11

Combine fractions.

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Step 11.1

Move the negative in front of the fraction.

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

Step 11.2

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

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

Step 11.3

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Combine fractions.

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Step 15.1

Add $1$ and $0$ .

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

Step 15.2

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

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

Step 15.3

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

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

Step 16

Simplify.

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Step 16.1

Apply the distributive property.

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

Step 16.2

Apply the distributive property.

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

Step 16.3

Apply the distributive property.

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

Step 16.4

Simplify the numerator.

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Step 16.4.1

Factor $3$ out of $3 {(x + 3)}^{\frac{1}{2}} + 3 (- x - 1 \cdot 2) \frac{1}{2 {(x + 3)}^{\frac{1}{2}}}$ .

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Step 16.4.1.1

Factor $3$ out of $3 {(x + 3)}^{\frac{1}{2}}$ .

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

Step 16.4.1.2

Factor $3$ out of $3 (- x - 1 \cdot 2) \frac{1}{2 {(x + 3)}^{\frac{1}{2}}}$ .

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

Step 16.4.1.3

Factor $3$ out of $3 ({(x + 3)}^{\frac{1}{2}}) + 3 ((- x - 1 \cdot 2) \frac{1}{2 {(x + 3)}^{\frac{1}{2}}})$ .

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

Step 16.4.2

Let $u_{2} = {(x + 3)}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(x + 3)}^{\frac{1}{2}}$ .

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Step 16.4.2.1

Rewrite using the commutative property of multiplication.

$\frac{3 \frac{2 u_{2} \cdot u_{2} - x - 2}{2 u_{2}}}{2 x + 2 \cdot 3}$

Step 16.4.2.2

Multiply $u_{2}$ by $u_{2}$ by adding the exponents.

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Step 16.4.2.2.1

Move $u_{2}$ .

$\frac{3 \frac{2 (u_{2} \cdot u_{2}) - x - 2}{2 u_{2}}}{2 x + 2 \cdot 3}$

Step 16.4.2.2.2

Multiply $u_{2}$ by $u_{2}$ .

$\frac{3 \frac{2 {u_{2}}^{2} - x - 2}{2 u_{2}}}{2 x + 2 \cdot 3}$

Step 16.4.3

Replace all occurrences of $u_{2}$ with ${(x + 3)}^{\frac{1}{2}}$ .

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

Step 16.4.4

Simplify.

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Step 16.4.4.1

Simplify each term.

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Step 16.4.4.1.1

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

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Step 16.4.4.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 16.4.4.1.1.2

Cancel the common factor of $2$ .

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Step 16.4.4.1.1.2.1

Cancel the common factor.

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

Step 16.4.4.1.1.2.2

Rewrite the expression.

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

Step 16.4.4.1.2

Simplify.

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

Step 16.4.4.1.3

Apply the distributive property.

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

Step 16.4.4.1.4

Multiply $2$ by $3$ .

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

Step 16.4.4.2

Subtract $x$ from $2 x$ .

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

Step 16.4.4.3

Subtract $2$ from $6$ .

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

Step 16.5

Combine terms.

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Step 16.5.1

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

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

Step 16.5.2

Multiply $2$ by $3$ .

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

Step 16.5.3

Rewrite $\frac{\frac{3 (x + 4)}{2 {(x + 3)}^{\frac{1}{2}}}}{2 x + 6}$ as a product.

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

Step 16.5.4

Multiply $\frac{3 (x + 4)}{2 {(x + 3)}^{\frac{1}{2}}}$ by $\frac{1}{2 x + 6}$ .

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

Step 16.6

Simplify the denominator.

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Step 16.6.1

Factor $2$ out of $2 x + 6$ .

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Step 16.6.1.1

Factor $2$ out of $2 x$ .

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

Step 16.6.1.2

Factor $2$ out of $6$ .

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

Step 16.6.1.3

Factor $2$ out of $2 x + 2 \cdot 3$ .

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

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

Step 16.6.2

Combine exponents.

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Step 16.6.2.1

Multiply $2$ by $2$ .

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

Step 16.6.2.2

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

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

Step 16.6.2.3

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

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

Step 16.6.2.4

Write $1$ as a fraction with a common denominator.

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

Step 16.6.2.5

Combine the numerators over the common denominator.

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

Step 16.6.2.6

Add $2$ and $1$ .

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