Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x + 1$ .

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Step 3.1

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

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

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

Step 3.3

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $1$ and $0$ .

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

Step 4.4.2

Multiply $3$ by $1$ .

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

Step 4.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

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

Step 8.2

Subtract $2$ from $3$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Combine the numerators over the common denominator.

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

Step 14

Multiply $2$ by $3$ .

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

Step 15

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

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

Step 16

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

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

Step 17

Simplify.

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Step 17.1

Simplify the numerator.

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Step 17.1.1

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

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Step 17.1.1.1

Reorder $x^{\frac{3}{2}}$ and $6$ .

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

Step 17.1.1.2

Factor $x^{\frac{1}{2}} \cdot 3 {(x + 1)}^{2}$ out of $6 \cdot x^{\frac{3}{2}} {(x + 1)}^{2}$ .

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

Step 17.1.1.3

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

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

Step 17.1.1.4

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

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

Step 17.1.2

Divide $2$ by $2$ .

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

Step 17.1.3

Simplify.

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

Step 17.1.4

Apply the distributive property.

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

Step 17.1.5

Multiply $- 1$ by $1$ .

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

Step 17.1.6

Subtract $x$ from $2 x$ .

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

Step 17.2

Combine terms.

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Step 17.2.1

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

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

Step 17.2.2

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

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

Step 17.2.3

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

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Step 17.2.3.1

Move $x^{- \frac{1}{2}}$ .

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

Step 17.2.3.2

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

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

Step 17.2.3.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 17.2.3.4

Combine $3$ and $\frac{2}{2}$ .

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

Step 17.2.3.5

Combine the numerators over the common denominator.

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

Step 17.2.3.6

Simplify the numerator.

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Step 17.2.3.6.1

Multiply $3$ by $2$ .

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

Step 17.2.3.6.2

Add $- 1$ and $6$ .

$\frac{3 {(x + 1)}^{2} (x - 1)}{2 x^{\frac{5}{2}}}$