## 17Calculus Derivative - Constant, Constant Multiple and Addition and Subtraction Rules

This page covers the first three basic rules when taking derivatives, the constant rule, constant multiple rule and the addition/subtraction rule.

Constant Rule

The constant rule is the simplest and most easily understood rule. The derivative calculates the slope, right? So, if you are given a horizontal line, what is the slope? Right! The slope is zero. That's it. That's the slope of every horizontal line. We can write the equation of a horizontal line as $$f(x)=c$$ where $$c$$ is a real number. Since these are always horizontal lines, the slope is zero. Therefore, the derivative of all constant functions (horizontal lines) is zero. We can derive this idea from the limit definition as follows. If $$f(x)=c$$ $f~'(x) = \lim_{h \to 0}{\frac{f(x+h) - f(x)}{h}} = \lim_{h \to 0}{\frac{c - c}{h}} = \lim_{h \to 0}{0} = 0$ Notice that c is gone from the final answer, $$f~'(x)=0$$, so this holds for all horizontal lines. Thinking about this, it makes sense intuitively, right?

Constant Multiple Rule

This rule works as you would expect. Mathematically, it looks like this. $\frac{d}{dx}[cf(x)] = c \frac{d}{dx}[f(x)]$ Nothing surprising, just pull out the constant and take the derivative of the function. This is discussed in more detail with examples on the power rule page.

When you have two functions that are added or subtracted, you just take the derivative of each individually. Mathematically, it looks like this. $\frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)]$ Nothing surprising or tricky here. It works just as you would expect.
[However, you will find out soon that this idea does NOT hold for multiplication and division. We have some special rules for those called the product rule and quotient rule.]

You will get plenty of chances to practice these techniques on the next page discussing the power rule.

You CAN Ace Calculus

### Calculus Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

 The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.
 Constant Rule Constant Multiple Rule Addition and Subtraction Rules

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.