Precalculus Examples

$2 y = - 3 x$

Step 1

Write $y = - \frac{3 x}{2}$ as a function.

$f (x) = - \frac{3 x}{2}$

Step 2

Divide each term in $2 y = - 3 x$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 y = - 3 x$ by $2$ .

$\frac{2 y}{2} = \frac{- 3 x}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 3 x}{2}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3 x}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$y = - \frac{3 x}{2}$

Step 3

Consider the difference quotient formula.

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

Step 4

Find the components of the definition.

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Step 4.1

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

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Step 4.1.1

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

$f (x + h) = - \frac{3 (x + h)}{2}$

Step 4.1.2

The final answer is $- \frac{3 (x + h)}{2}$ .

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

Step 4.2

Find the components of the definition.

$f (x + h) = - \frac{3 (x + h)}{2}$

$f (x) = - \frac{3 x}{2}$

$f (x + h) = - \frac{3 (x + h)}{2}$

$f (x) = - \frac{3 x}{2}$

Step 5

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{3 (x + h)}{2} - (- \frac{3 x}{2})}{h}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $- - \frac{3 x}{2}$ .

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Step 6.1.1.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.1.2

Multiply $\frac{3 x}{2}$ by $1$ .

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

Step 6.1.2

Combine the numerators over the common denominator.

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

Step 6.1.3

Rewrite $\frac{- 3 (x + h) + 3 x}{2}$ in a factored form.

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Step 6.1.3.1

Factor $3$ out of $- 3 (x + h) + 3 x$ .

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Step 6.1.3.1.1

Factor $3$ out of $- 3 (x + h)$ .

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

Step 6.1.3.1.2

Factor $3$ out of $3 x$ .

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

Step 6.1.3.1.3

Factor $3$ out of $3 (- (x + h)) + 3 (x)$ .

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

Step 6.1.3.2

Apply the distributive property.

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

Step 6.1.3.3

Add $- x$ and $x$ .

$\frac{\frac{3 (- h + 0)}{2}}{h}$

Step 6.1.3.4

Add $- h$ and $0$ .

$\frac{\frac{3 \cdot - 1 h}{2}}{h}$

Step 6.1.3.5

Combine exponents.

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Step 6.1.3.5.1

Factor out negative.

$\frac{\frac{- (3 h)}{2}}{h}$

Step 6.1.3.5.2

Multiply $3$ by $- 1$ .

$\frac{\frac{- 3 h}{2}}{h}$

Step 6.1.4

Move the negative in front of the fraction.

$\frac{- \frac{3 h}{2}}{h}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{3 h}{2} \cdot \frac{1}{h}$

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Move the leading negative in $- \frac{3 h}{2}$ into the numerator.

$\frac{- 3 h}{2} \cdot \frac{1}{h}$

Step 6.3.2

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

$\frac{h \cdot - 3}{2} \cdot \frac{1}{h}$

Step 6.3.3

Cancel the common factor.

$\frac{h \cdot - 3}{2} \cdot \frac{1}{h}$

Step 6.3.4

Rewrite the expression.

$\frac{- 3}{2}$

Step 6.4

Move the negative in front of the fraction.

$- \frac{3}{2}$

Step 7