The first two sections of this module introduce you to the formal definition of a derivative which involves a large amount of algebra. Here we will review evaluating a function at variable inputs, including how to find a function’s difference quotient. Since writing the equation of a tangent line is required in these sections, writing the equation of a line will also be reviewed. Finally, a refresher about how to rationalize and simplify complex rational expressions will be given.

Evaluate Functions at Variable Inputs

You likely have plenty of experience evaluating functions at constants. Functions can also be evaluated for inputs that are variables or expressions. The process is the same, but the simplified answer will contain a variable. The following example shows you how to evaluate a function for a variable input.

In the following video, we show more examples of evaluating functions for both constant and variable inputs.

You can view the transcript for “Ex: Determine Various Function Outputs for a Quadratic Function” here (opens in new window).

Write the Equation of a Line

To write the equation of a line, the line’s slope and a point the line goes through must be known. Perhaps the most familiar form of a linear equation is slope-intercept form written as [latex]y=mx+b[/latex], where [latex]m=\text{slope}[/latex] and [latex]b=y\text{-intercept}[/latex]. Let us begin with the slope.

Often, the starting point to writing the equation of a line is to use point-slope formula. Given the slope and one point on a line, we can find the equation of the line using point-slope form shown below.

We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.

Rationalize Radical Expressions

(also in Module 2, Skills Review for The Limit Laws)

When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator.

We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by a form of 1 that will eliminate the radical.

For a denominator containing a binomial where at least one of the terms is a square root, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign in the middle of the binomial. If the denominator is [latex]a+b\sqrt{c}[/latex], then the conjugate is [latex]a-b\sqrt{c}[/latex].

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression [latex]\dfrac{a}{\dfrac{1}{b}+c}[/latex] can be simplified by rewriting the numerator as the fraction [latex]\dfrac{a}{1}[/latex] and combining the expressions in the denominator as [latex]\dfrac{1+bc}{b}[/latex]. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get [latex]\dfrac{a}{1}\cdot \dfrac{b}{1+bc}[/latex] which is equal to [latex]\dfrac{ab}{1+bc}[/latex].