Calculus Examples

$y = \frac{8 x^{\frac{3}{2}}}{x}$

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 $f (x) = 8 x^{\frac{3}{2}}$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [8 x^{\frac{3}{2}}] - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 2

Differentiate.

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Step 2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{\frac{3}{2}}$ with respect to $x$ is $8 \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$\frac{x (8 \frac{d}{d x} [x^{\frac{3}{2}}]) - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 2.2

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

$\frac{x (8 (\frac{3}{2} x^{\frac{3}{2} - 1})) - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 3

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

$\frac{x (8 (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})) - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 4

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

$\frac{x (8 (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})) - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 5

Combine the numerators over the common denominator.

$\frac{x (8 (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})) - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$\frac{x (8 (\frac{3}{2} x^{\frac{3 - 2}{2}})) - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 6.2

Subtract $2$ from $3$ .

$\frac{x (8 (\frac{3}{2} x^{\frac{1}{2}})) - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 7

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

$\frac{x (8 \frac{3 x^{\frac{1}{2}}}{2}) - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 8

Combine $8$ and $\frac{3 x^{\frac{1}{2}}}{2}$ .

$\frac{x \frac{8 (3 x^{\frac{1}{2}})}{2} - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 9

Multiply $3$ by $8$ .

$\frac{x \frac{24 x^{\frac{1}{2}}}{2} - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 10

Factor $2$ out of $24 x^{\frac{1}{2}}$ .

$\frac{x \frac{2 (12 x^{\frac{1}{2}})}{2} - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 11

Cancel the common factors.

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Step 11.1

Factor $2$ out of $2$ .

$\frac{x \frac{2 (12 x^{\frac{1}{2}})}{2 (1)} - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 11.2

Cancel the common factor.

$\frac{x \frac{2 (12 x^{\frac{1}{2}})}{2 \cdot 1} - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 11.3

Rewrite the expression.

$\frac{x \frac{12 x^{\frac{1}{2}}}{1} - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 11.4

Divide $12 x^{\frac{1}{2}}$ by $1$ .

$\frac{x (12 x^{\frac{1}{2}}) - 1 \cdot 8 x^{\frac{3}{2}} \frac{d}{d x} [x]}{x^{2}}$

Step 12

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

$\frac{x (12 x^{\frac{1}{2}}) - 1 \cdot 8 x^{\frac{3}{2}} \cdot 1}{x^{2}}$

Step 13

Simplify.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Simplify each term.

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Step 13.1.1.1

Rewrite using the commutative property of multiplication.

$\frac{12 x \cdot x^{\frac{1}{2}} - 1 \cdot 8 x^{\frac{3}{2}} \cdot 1}{x^{2}}$

Step 13.1.1.2

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

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Step 13.1.1.2.1

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

$\frac{12 (x^{\frac{1}{2}} x) - 1 \cdot 8 x^{\frac{3}{2}} \cdot 1}{x^{2}}$

Step 13.1.1.2.2

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

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Step 13.1.1.2.2.1

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

$\frac{12 (x^{\frac{1}{2}} x^{1}) - 1 \cdot 8 x^{\frac{3}{2}} \cdot 1}{x^{2}}$

Step 13.1.1.2.2.2

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

$\frac{12 x^{\frac{1}{2} + 1} - 1 \cdot 8 x^{\frac{3}{2}} \cdot 1}{x^{2}}$

Step 13.1.1.2.3

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

$\frac{12 x^{\frac{1}{2} + \frac{2}{2}} - 1 \cdot 8 x^{\frac{3}{2}} \cdot 1}{x^{2}}$

Step 13.1.1.2.4

Combine the numerators over the common denominator.

$\frac{12 x^{\frac{1 + 2}{2}} - 1 \cdot 8 x^{\frac{3}{2}} \cdot 1}{x^{2}}$

Step 13.1.1.2.5

Add $1$ and $2$ .

$\frac{12 x^{\frac{3}{2}} - 1 \cdot 8 x^{\frac{3}{2}} \cdot 1}{x^{2}}$

Step 13.1.1.3

Multiply $- 1$ by $8$ .

$\frac{12 x^{\frac{3}{2}} - 8 x^{\frac{3}{2}} \cdot 1}{x^{2}}$

Step 13.1.1.4

Multiply $- 8$ by $1$ .

$\frac{12 x^{\frac{3}{2}} - 8 x^{\frac{3}{2}}}{x^{2}}$

Step 13.1.2

Subtract $8 x^{\frac{3}{2}}$ from $12 x^{\frac{3}{2}}$ .

$\frac{4 x^{\frac{3}{2}}}{x^{2}}$

Step 13.2

Combine terms.

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Step 13.2.1

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

$\frac{4}{x^{2} x^{- \frac{3}{2}}}$

Step 13.2.2

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

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Step 13.2.2.1

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

$\frac{4}{x^{2 - \frac{3}{2}}}$

Step 13.2.2.2

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

$\frac{4}{x^{2 \cdot \frac{2}{2} - \frac{3}{2}}}$

Step 13.2.2.3

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

$\frac{4}{x^{\frac{2 \cdot 2}{2} - \frac{3}{2}}}$

Step 13.2.2.4

Combine the numerators over the common denominator.

$\frac{4}{x^{\frac{2 \cdot 2 - 3}{2}}}$

Step 13.2.2.5

Simplify the numerator.

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Step 13.2.2.5.1

Multiply $2$ by $2$ .

$\frac{4}{x^{\frac{4 - 3}{2}}}$

Step 13.2.2.5.2

Subtract $3$ from $4$ .

$\frac{4}{x^{\frac{1}{2}}}$