Precalculus Examples

$f (x) = 0.03 (10 x - x^{2})$

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.

$f (x + h) = 0.03 (10 (x + h) - {(x + h)}^{2})$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Apply the distributive property.

$f (x + h) = 0.03 (10 x + 10 h - {(x + h)}^{2})$

Step 2.1.2.1.2

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

$f (x + h) = 0.03 (10 x + 10 h - ((x + h) (x + h)))$

Step 2.1.2.1.3

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 2.1.2.1.3.1

Apply the distributive property.

$f (x + h) = 0.03 (10 x + 10 h - (x (x + h) + h (x + h)))$

Step 2.1.2.1.3.2

Apply the distributive property.

$f (x + h) = 0.03 (10 x + 10 h - (x \cdot x + x h + h (x + h)))$

Step 2.1.2.1.3.3

Apply the distributive property.

$f (x + h) = 0.03 (10 x + 10 h - (x \cdot x + x h + h x + h \cdot h))$

Step 2.1.2.1.4

Simplify and combine like terms.

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Step 2.1.2.1.4.1

Simplify each term.

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Step 2.1.2.1.4.1.1

Multiply $x$ by $x$ .

$f (x + h) = 0.03 (10 x + 10 h - (x^{2} + x h + h x + h \cdot h))$

Step 2.1.2.1.4.1.2

Multiply $h$ by $h$ .

$f (x + h) = 0.03 (10 x + 10 h - (x^{2} + x h + h x + h^{2}))$

Step 2.1.2.1.4.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.4.2.1

Reorder $x$ and $h$ .

$f (x + h) = 0.03 (10 x + 10 h - (x^{2} + h x + h x + h^{2}))$

Step 2.1.2.1.4.2.2

Add $h x$ and $h x$ .

$f (x + h) = 0.03 (10 x + 10 h - (x^{2} + 2 h x + h^{2}))$

Step 2.1.2.1.5

Apply the distributive property.

$f (x + h) = 0.03 (10 x + 10 h - x^{2} - (2 h x) - h^{2})$

Step 2.1.2.1.6

Multiply $2$ by $- 1$ .

$f (x + h) = 0.03 (10 x + 10 h - x^{2} - 2 h x - h^{2})$

Step 2.1.2.2

Apply the distributive property.

$f (x + h) = 0.03 (10 x) + 0.03 (10 h) + 0.03 (- x^{2}) + 0.03 (- 2 h x) + 0.03 (- h^{2})$

Step 2.1.2.3

Simplify.

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Step 2.1.2.3.1

Multiply $10$ by $0.03$ .

$f (x + h) = 0.3 x + 0.03 (10 h) + 0.03 (- x^{2}) + 0.03 (- 2 h x) + 0.03 (- h^{2})$

Step 2.1.2.3.2

Multiply $10$ by $0.03$ .

$f (x + h) = 0.3 x + 0.3 h + 0.03 (- x^{2}) + 0.03 (- 2 h x) + 0.03 (- h^{2})$

Step 2.1.2.3.3

Multiply $- 1$ by $0.03$ .

$f (x + h) = 0.3 x + 0.3 h - 0.03 x^{2} + 0.03 (- 2 h x) + 0.03 (- h^{2})$

Step 2.1.2.3.4

Multiply $- 2$ by $0.03$ .

$f (x + h) = 0.3 x + 0.3 h - 0.03 x^{2} - 0.06 (h x) + 0.03 (- h^{2})$

Step 2.1.2.3.5

Multiply $- 1$ by $0.03$ .

$f (x + h) = 0.3 x + 0.3 h - 0.03 x^{2} - 0.06 (h x) - 0.03 h^{2}$

$f (x + h) = 0.3 x + 0.3 h - 0.03 x^{2} - 0.06 h x - 0.03 h^{2}$

Step 2.1.2.4

The final answer is $0.3 x + 0.3 h - 0.03 x^{2} - 0.06 h x - 0.03 h^{2}$ .

$0.3 x + 0.3 h - 0.03 x^{2} - 0.06 h x - 0.03 h^{2}$

Step 2.2

Reorder.

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Step 2.2.1

Move $0.3 x$ .

$0.3 h - 0.03 x^{2} - 0.06 h x - 0.03 h^{2} + 0.3 x$

Step 2.2.2

Move $0.3 h$ .

$- 0.03 x^{2} - 0.06 h x - 0.03 h^{2} + 0.3 h + 0.3 x$

Step 2.2.3

Move $- 0.03 x^{2}$ .

$- 0.06 h x - 0.03 h^{2} - 0.03 x^{2} + 0.3 h + 0.3 x$

Step 2.2.4

Reorder $- 0.06 h x$ and $- 0.03 h^{2}$ .

$- 0.03 h^{2} - 0.06 h x - 0.03 x^{2} + 0.3 h + 0.3 x$

Step 2.3

Find the components of the definition.

$f (x + h) = - 0.03 h^{2} - 0.06 h x - 0.03 x^{2} + 0.3 h + 0.3 x$

$f (x) = 0.3 x - 0.03 x^{2}$

$f (x + h) = - 0.03 h^{2} - 0.06 h x - 0.03 x^{2} + 0.3 h + 0.3 x$

$f (x) = 0.3 x - 0.03 x^{2}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 0.03 h^{2} - 0.06 h x - 0.03 x^{2} + 0.3 h + 0.3 x - (0.3 x - 0.03 x^{2})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $0.01$ out of $- 0.03 h^{2} - 0.06 h x - 0.03 x^{2} + 0.3 h + 0.3 x - (0.3 x - 0.03 x^{2})$ .

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Step 4.1.1.1

Reorder the expression.

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Step 4.1.1.1.1

Move $h$ .

$\frac{- 0.03 h^{2} - 0.06 x h - 0.03 x^{2} + 0.3 h + 0.3 x - (0.3 x - 0.03 x^{2})}{h}$

Step 4.1.1.1.2

Reorder $0.3 x$ and $- 0.03 x^{2}$ .

$\frac{- 0.03 h^{2} - 0.06 x h - 0.03 x^{2} + 0.3 h + 0.3 x - (- 0.03 x^{2} + 0.3 x)}{h}$

Step 4.1.1.2

Factor $0.01$ out of $- 0.03 h^{2}$ .

$\frac{0.01 (- 3 h^{2}) - 0.06 x h - 0.03 x^{2} + 0.3 h + 0.3 x - (- 0.03 x^{2} + 0.3 x)}{h}$

Step 4.1.1.3

Factor $0.01$ out of $- 0.06 x h$ .

$\frac{0.01 (- 3 h^{2}) + 0.01 (- 6 x h) - 0.03 x^{2} + 0.3 h + 0.3 x - (- 0.03 x^{2} + 0.3 x)}{h}$

Step 4.1.1.4

Factor $0.01$ out of $- 0.03 x^{2}$ .

$\frac{0.01 (- 3 h^{2}) + 0.01 (- 6 x h) + 0.01 (- 3 x^{2}) + 0.3 h + 0.3 x - (- 0.03 x^{2} + 0.3 x)}{h}$

Step 4.1.1.5

Factor $0.01$ out of $0.3 h$ .

$\frac{0.01 (- 3 h^{2}) + 0.01 (- 6 x h) + 0.01 (- 3 x^{2}) + 0.01 (30 h) + 0.3 x - (- 0.03 x^{2} + 0.3 x)}{h}$

Step 4.1.1.6

Factor $0.01$ out of $0.3 x$ .

$\frac{0.01 (- 3 h^{2}) + 0.01 (- 6 x h) + 0.01 (- 3 x^{2}) + 0.01 (30 h) + 0.01 (30 x) - (- 0.03 x^{2} + 0.3 x)}{h}$

Step 4.1.1.7

Factor $0.01$ out of $- (- 0.03 x^{2} + 0.3 x)$ .

$\frac{0.01 (- 3 h^{2}) + 0.01 (- 6 x h) + 0.01 (- 3 x^{2}) + 0.01 (30 h) + 0.01 (30 x) + 0.01 (- 100 (- 0.03 x^{2} + 0.3 x))}{h}$

Step 4.1.1.8

Factor $0.01$ out of $0.01 (- 3 h^{2}) + 0.01 (- 6 x h)$ .

$\frac{0.01 (- 3 h^{2} - 6 x h) + 0.01 (- 3 x^{2}) + 0.01 (30 h) + 0.01 (30 x) + 0.01 (- 100 (- 0.03 x^{2} + 0.3 x))}{h}$

Step 4.1.1.9

Factor $0.01$ out of $0.01 (- 3 h^{2} - 6 x h) + 0.01 (- 3 x^{2})$ .

$\frac{0.01 (- 3 h^{2} - 6 x h - 3 x^{2}) + 0.01 (30 h) + 0.01 (30 x) + 0.01 (- 100 (- 0.03 x^{2} + 0.3 x))}{h}$

Step 4.1.1.10

Factor $0.01$ out of $0.01 (- 3 h^{2} - 6 x h - 3 x^{2}) + 0.01 (30 h)$ .

$\frac{0.01 (- 3 h^{2} - 6 x h - 3 x^{2} + 30 h) + 0.01 (30 x) + 0.01 (- 100 (- 0.03 x^{2} + 0.3 x))}{h}$

Step 4.1.1.11

Factor $0.01$ out of $0.01 (- 3 h^{2} - 6 x h - 3 x^{2} + 30 h) + 0.01 (30 x)$ .

$\frac{0.01 (- 3 h^{2} - 6 x h - 3 x^{2} + 30 h + 30 x) + 0.01 (- 100 (- 0.03 x^{2} + 0.3 x))}{h}$

Step 4.1.1.12

Factor $0.01$ out of $0.01 (- 3 h^{2} - 6 x h - 3 x^{2} + 30 h + 30 x) + 0.01 (- 100 (- 0.03 x^{2} + 0.3 x))$ .

$\frac{0.01 (- 3 h^{2} - 6 x h - 3 x^{2} + 30 h + 30 x - 100 (- 0.03 x^{2} + 0.3 x))}{h}$

Step 4.1.2

Apply the distributive property.

$\frac{0.01 (- 3 h^{2} - 6 x h - 3 x^{2} + 30 h + 30 x - 100 (- 0.03 x^{2}) - 100 (0.3 x))}{h}$

Step 4.1.3

Multiply $- 0.03$ by $- 100$ .

$\frac{0.01 (- 3 h^{2} - 6 x h - 3 x^{2} + 30 h + 30 x + 3 x^{2} - 100 (0.3 x))}{h}$

Step 4.1.4

Multiply $0.3$ by $- 100$ .

$\frac{0.01 (- 3 h^{2} - 6 x h - 3 x^{2} + 30 h + 30 x + 3 x^{2} - 30 x)}{h}$

Step 4.1.5

Add $- 3 x^{2}$ and $3 x^{2}$ .

$\frac{0.01 (- 3 h^{2} - 6 x h + 0 x^{2} + 30 h + 30 x - 30 x)}{h}$

Step 4.1.6

Subtract $30 x$ from $30 x$ .

$\frac{0.01 (- 3 h^{2} - 6 x h + 0 x^{2} + 30 h + 0 x)}{h}$

Step 4.1.7

Rewrite $- 3 h^{2} - 6 x h + 0 x^{2} + 30 h + 0 x$ in a factored form.

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Step 4.1.7.1

Factor $3$ out of $- 3 h^{2} - 6 x h + 0 x^{2} + 30 h + 0 x$ .

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Step 4.1.7.1.1

Factor $3$ out of $- 3 h^{2}$ .

$\frac{0.01 (3 (- h^{2}) - 6 x h + 0 x^{2} + 30 h + 0 x)}{h}$

Step 4.1.7.1.2

Factor $3$ out of $- 6 x h$ .

$\frac{0.01 (3 (- h^{2}) + 3 (- 2 x h) + 0 x^{2} + 30 h + 0 x)}{h}$

Step 4.1.7.1.3

Factor $3$ out of $0 x^{2}$ .

$\frac{0.01 (3 (- h^{2}) + 3 (- 2 x h) + 3 (0 x^{2}) + 30 h + 0 x)}{h}$

Step 4.1.7.1.4

Factor $3$ out of $30 h$ .

$\frac{0.01 (3 (- h^{2}) + 3 (- 2 x h) + 3 (0 x^{2}) + 3 (10 h) + 0 x)}{h}$

Step 4.1.7.1.5

Factor $3$ out of $0 x$ .

$\frac{0.01 (3 (- h^{2}) + 3 (- 2 x h) + 3 (0 x^{2}) + 3 (10 h) + 3 (0 x))}{h}$

Step 4.1.7.1.6

Factor $3$ out of $3 (- h^{2}) + 3 (- 2 x h)$ .

$\frac{0.01 (3 (- h^{2} - 2 x h) + 3 (0 x^{2}) + 3 (10 h) + 3 (0 x))}{h}$

Step 4.1.7.1.7

Factor $3$ out of $3 (- h^{2} - 2 x h) + 3 (0 x^{2})$ .

$\frac{0.01 (3 (- h^{2} - 2 x h + 0 x^{2}) + 3 (10 h) + 3 (0 x))}{h}$

Step 4.1.7.1.8

Factor $3$ out of $3 (- h^{2} - 2 x h + 0 x^{2}) + 3 (10 h)$ .

$\frac{0.01 (3 (- h^{2} - 2 x h + 0 x^{2} + 10 h) + 3 (0 x))}{h}$

Step 4.1.7.1.9

Factor $3$ out of $3 (- h^{2} - 2 x h + 0 x^{2} + 10 h) + 3 (0 x)$ .

$\frac{0.01 (3 (- h^{2} - 2 x h + 0 x^{2} + 10 h + 0 x))}{h}$

Step 4.1.7.2

Multiply $0$ by $x^{2}$ .

$\frac{0.01 (3 (- h^{2} - 2 x h + 0 + 10 h + 0 x))}{h}$

Step 4.1.7.3

Multiply $0$ by $x$ .

$\frac{0.01 (3 (- h^{2} - 2 x h + 0 + 10 h + 0))}{h}$

Step 4.1.7.4

Add $- h^{2} - 2 x h + 0 + 10 h$ and $0$ .

$\frac{0.01 (3 (- h^{2} - 2 x h + 0 + 10 h))}{h}$

Step 4.1.7.5

Add $- h^{2}$ and $0$ .

$\frac{0.01 (3 (- h^{2} - 2 x h + 10 h))}{h}$

Step 4.1.7.6

Factor $h$ out of $- h^{2} - 2 x h + 10 h$ .

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Step 4.1.7.6.1

Factor $h$ out of $- h^{2}$ .

$\frac{0.01 (3 (h (- h) - 2 x h + 10 h))}{h}$

Step 4.1.7.6.2

Factor $h$ out of $- 2 x h$ .

$\frac{0.01 (3 (h (- h) + h (- 2 x) + 10 h))}{h}$

Step 4.1.7.6.3

Factor $h$ out of $10 h$ .

$\frac{0.01 (3 (h (- h) + h (- 2 x) + h \cdot 10))}{h}$

Step 4.1.7.6.4

Factor $h$ out of $h (- h) + h (- 2 x)$ .

$\frac{0.01 (3 (h (- h - 2 x) + h \cdot 10))}{h}$

Step 4.1.7.6.5

Factor $h$ out of $h (- h - 2 x) + h \cdot 10$ .

$\frac{0.01 (3 (h (- h - 2 x + 10)))}{h}$

$\frac{0.01 (3 h (- h - 2 x + 10))}{h}$

$\frac{0.01 \cdot 3 h (- h - 2 x + 10)}{h}$

Step 4.1.8

Multiply $0.01$ by $3$ .

$\frac{0.03 h (- h - 2 x + 10)}{h}$

Step 4.2

Simplify terms.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.03 h (- h - 2 x + 10)}{h}$

Step 4.2.1.2

Divide $0.03 (- h - 2 x + 10)$ by $1$ .

$0.03 (- h - 2 x + 10)$

Step 4.2.2

Apply the distributive property.

$0.03 (- h) + 0.03 (- 2 x) + 0.03 \cdot 10$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $- 1$ by $0.03$ .

$- 0.03 h + 0.03 (- 2 x) + 0.03 \cdot 10$

Step 4.3.2

Multiply $- 2$ by $0.03$ .

$- 0.03 h - 0.06 x + 0.03 \cdot 10$

Step 4.3.3

Multiply $0.03$ by $10$ .

$- 0.03 h - 0.06 x + 0.3$

Step 4.4

Reorder $- 0.03 h$ and $- 0.06 x$ .

$- 0.06 x - 0.03 h + 0.3$

Step 5