Precalculus Examples

$R (x) = 45 x^{\frac{9}{10}}$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$R (x + h) = 45 {(x + h)}^{\frac{9}{10}}$

Step 2.1.2

The final answer is $45 {(x + h)}^{\frac{9}{10}}$ .

$45 {(x + h)}^{\frac{9}{10}}$

Step 2.2

Find the components of the definition.

$R (x + h) = 45 {(x + h)}^{\frac{9}{10}}$

$R (x) = 45 x^{\frac{9}{10}}$

$R (x + h) = 45 {(x + h)}^{\frac{9}{10}}$

$R (x) = 45 x^{\frac{9}{10}}$

Step 3

Plug in the components.

$\frac{R (x + h) - R x}{h} = \frac{45 {(x + h)}^{\frac{9}{10}} - (45 x^{\frac{9}{10}})}{h}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $- 1$ by $45$ .

$\frac{45 {(x + h)}^{\frac{9}{10}} - 45 x^{\frac{9}{10}}}{h}$

Step 4.2

Rewrite $45 {(x + h)}^{\frac{9}{10}} - 45 x^{\frac{9}{10}}$ in a factored form.

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Step 4.2.1

Factor $45$ out of $45 {(x + h)}^{\frac{9}{10}} - 45 x^{\frac{9}{10}}$ .

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Step 4.2.1.1

Factor $45$ out of $45 {(x + h)}^{\frac{9}{10}}$ .

$\frac{45 {(x + h)}^{\frac{9}{10}} - 45 x^{\frac{9}{10}}}{h}$

Step 4.2.1.2

Factor $45$ out of $- 45 x^{\frac{9}{10}}$ .

$\frac{45 {(x + h)}^{\frac{9}{10}} + 45 (- x^{\frac{9}{10}})}{h}$

Step 4.2.1.3

Factor $45$ out of $45 {(x + h)}^{\frac{9}{10}} + 45 (- x^{\frac{9}{10}})$ .

$\frac{45 ({(x + h)}^{\frac{9}{10}} - x^{\frac{9}{10}})}{h}$

Step 4.2.2

Rewrite ${(x + h)}^{\frac{9}{10}}$ as ${({(x + h)}^{\frac{3}{10}})}^{3}$ .

$\frac{45 ({({(x + h)}^{\frac{3}{10}})}^{3} - x^{\frac{9}{10}})}{h}$

Step 4.2.3

Rewrite $x^{\frac{9}{10}}$ as ${(x^{\frac{3}{10}})}^{3}$ .

$\frac{45 ({({(x + h)}^{\frac{3}{10}})}^{3} - {(x^{\frac{3}{10}})}^{3})}{h}$

Step 4.2.4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = {(x + h)}^{\frac{3}{10}}$ and $b = x^{\frac{3}{10}}$ .

$\frac{45 (({(x + h)}^{\frac{3}{10}} - x^{\frac{3}{10}}) ({({(x + h)}^{\frac{3}{10}})}^{2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5

Simplify.

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Step 4.2.5.1

Rewrite ${(x + h)}^{\frac{3}{10}}$ as ${({(x + h)}^{\frac{1}{10}})}^{3}$ .

$\frac{45 (({({(x + h)}^{\frac{1}{10}})}^{3} - x^{\frac{3}{10}}) ({({(x + h)}^{\frac{3}{10}})}^{2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.2

Rewrite $x^{\frac{3}{10}}$ as ${(x^{\frac{1}{10}})}^{3}$ .

$\frac{45 (({({(x + h)}^{\frac{1}{10}})}^{3} - {(x^{\frac{1}{10}})}^{3}) ({({(x + h)}^{\frac{3}{10}})}^{2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = {(x + h)}^{\frac{1}{10}}$ and $b = x^{\frac{1}{10}}$ .

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({({(x + h)}^{\frac{1}{10}})}^{2} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + {(x^{\frac{1}{10}})}^{2}) ({({(x + h)}^{\frac{3}{10}})}^{2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.4

Simplify.

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Step 4.2.5.4.1

Multiply the exponents in ${({(x + h)}^{\frac{1}{10}})}^{2}$ .

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Step 4.2.5.4.1.1

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

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{10} \cdot 2} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + {(x^{\frac{1}{10}})}^{2}) ({({(x + h)}^{\frac{3}{10}})}^{2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.4.1.2

Cancel the common factor of $2$ .

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Step 4.2.5.4.1.2.1

Factor $2$ out of $10$ .

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{2 (5)} \cdot 2} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + {(x^{\frac{1}{10}})}^{2}) ({({(x + h)}^{\frac{3}{10}})}^{2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.4.1.2.2

Cancel the common factor.

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{2 \cdot 5} \cdot 2} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + {(x^{\frac{1}{10}})}^{2}) ({({(x + h)}^{\frac{3}{10}})}^{2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.4.1.2.3

Rewrite the expression.

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{5}} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + {(x^{\frac{1}{10}})}^{2}) ({({(x + h)}^{\frac{3}{10}})}^{2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.4.2

Multiply the exponents in ${(x^{\frac{1}{10}})}^{2}$ .

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Step 4.2.5.4.2.1

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

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{5}} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + x^{\frac{1}{10} \cdot 2}) ({({(x + h)}^{\frac{3}{10}})}^{2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.4.2.2

Cancel the common factor of $2$ .

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Step 4.2.5.4.2.2.1

Factor $2$ out of $10$ .

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{5}} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + x^{\frac{1}{2 (5)} \cdot 2}) ({({(x + h)}^{\frac{3}{10}})}^{2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.4.2.2.2

Cancel the common factor.

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{5}} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + x^{\frac{1}{2 \cdot 5} \cdot 2}) ({({(x + h)}^{\frac{3}{10}})}^{2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.4.2.2.3

Rewrite the expression.

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{5}} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + x^{\frac{1}{5}}) ({({(x + h)}^{\frac{3}{10}})}^{2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.5

Multiply the exponents in ${({(x + h)}^{\frac{3}{10}})}^{2}$ .

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Step 4.2.5.5.1

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

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{5}} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + x^{\frac{1}{5}}) ({(x + h)}^{\frac{3}{10} \cdot 2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.5.2

Cancel the common factor of $2$ .

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Step 4.2.5.5.2.1

Factor $2$ out of $10$ .

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{5}} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + x^{\frac{1}{5}}) ({(x + h)}^{\frac{3}{2 (5)} \cdot 2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.5.2.2

Cancel the common factor.

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{5}} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + x^{\frac{1}{5}}) ({(x + h)}^{\frac{3}{2 \cdot 5} \cdot 2} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.5.2.3

Rewrite the expression.

$\frac{45 (({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{5}} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + x^{\frac{1}{5}}) ({(x + h)}^{\frac{3}{5}} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + {(x^{\frac{3}{10}})}^{2}))}{h}$

Step 4.2.5.6

Multiply the exponents in ${(x^{\frac{3}{10}})}^{2}$ .

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Step 4.2.5.6.1

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

Step 4.2.5.6.2

Cancel the common factor of $2$ .

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Step 4.2.5.6.2.1

Factor $2$ out of $10$ .

Step 4.2.5.6.2.2

Cancel the common factor.

Step 4.2.5.6.2.3

Rewrite the expression.

$\frac{45 ({(x + h)}^{\frac{1}{10}} - x^{\frac{1}{10}}) ({(x + h)}^{\frac{1}{5}} + {(x + h)}^{\frac{1}{10}} x^{\frac{1}{10}} + x^{\frac{1}{5}}) ({(x + h)}^{\frac{3}{5}} + {(x + h)}^{\frac{3}{10}} x^{\frac{3}{10}} + x^{\frac{3}{5}})}{h}$

Step 5