Precalculus Examples

$2 y - 6 x = 0$

Step 1

Write $y = 3 x$ as a function.

$f (x) = 3 x$

Step 2

Solve for $y$ .

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Step 2.1

Add $6 x$ to both sides of the equation.

$2 y = 6 x$

Step 2.2

Divide each term in $2 y = 6 x$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 y = 6 x$ by $2$ .

$\frac{2 y}{2} = \frac{6 x}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{6 x}{2}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{6 x}{2}$

$y = \frac{6 x}{2}$

$y = \frac{6 x}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $6$ and $2$ .

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Step 2.2.3.1.1

Factor $2$ out of $6 x$ .

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

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $2$ out of $2$ .

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

Step 2.2.3.1.2.2

Cancel the common factor.

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

Step 2.2.3.1.2.3

Rewrite the expression.

$y = \frac{3 x}{1}$

Step 2.2.3.1.2.4

Divide $3 x$ by $1$ .

$y = 3 x$

$y = 3 x$

$y = 3 x$

$y = 3 x$

$y = 3 x$

$y = 3 x$

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.

$f (x + h) = 3 (x + h)$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Apply the distributive property.

$f (x + h) = 3 x + 3 h$

Step 4.1.2.2

The final answer is $3 x + 3 h$ .

$3 x + 3 h$

$3 x + 3 h$

$3 x + 3 h$

Step 4.2

Find the components of the definition.

$f (x + h) = 3 x + 3 h$

$f (x) = 3 x$

$f (x + h) = 3 x + 3 h$

$f (x) = 3 x$

Step 5

Plug in the components.

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $- 1$ by $3$ .

$\frac{3 x + 3 h - 3 x}{h}$

Step 6.1.2

Subtract $3 x$ from $3 x$ .

$\frac{3 h + 0}{h}$

Step 6.1.3

Add $3 h$ and $0$ .

$\frac{3 h}{h}$

$\frac{3 h}{h}$

Step 6.2

Cancel the common factor of $h$ .

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Step 6.2.1

Cancel the common factor.

$\frac{3 h}{h}$

Step 6.2.2

Divide $3$ by $1$ .

$3$

$3$

$3$

Step 7

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$