Derivatives
This page consists of some basic topics to get you started with derivatives. The practice problems use only the techniques discussed on this page. For additional derivative topics and techniques, see the menu at the top of the page.

Derivative Definition and Meaning
\(\displaystyle{ f~'(x) = }\) \(\displaystyle{ \lim_{\Delta x \to 0}{\frac{f(x+\Delta x) - f(x)}{\Delta x}} }\)
derivative = slope
you may also see the derivative referred to as the rate of change or instantaneous rate of change
Average vs Instantaneous Rate of Change Practice 1 Find the average rate of change of the function \(f(x)=x^4-5x\) over the interval \([0,3]\). solution Practice 1 Solution video by Krista King Math close solution

There are two very important things to remember about the derivative, the definition and what it means.
First, the definition of the derivative is from a limit. When you were studying limits, you may have run across this limit but not known what it meant. There are at least three different forms that you might see. These are sometimes called difference quotients.

Three Ways to Write the Limit Definition of the Derivative
\(\displaystyle{ f~'(x) = \lim_{\Delta x \to 0}{\frac{f(x+\Delta x) - f(x)}{\Delta x}} }\) original form; used the most since it contains the most information
\(\displaystyle{ f~'(x) = \lim_{h \to 0}{\frac{f(x+h) - f(x)}{h}} }\) alternative form of the first equation; some students find this form easier to use
\(\displaystyle{ f~'(a) = \lim_{x \to a}{\frac{f(x) - f(a)}{x-a}} }\) this form is usually used when finding the derivative at a point \(x=a\) but the first two are more common

The second thing you need to really know is that the derivative is the slope of the graph. Every time you hear the word 'derivative' you should think 'slope' and, when a problem asks for the slope (or the equation of the tangent line, since it involves slope) your first thought should be 'derivative'.

Okay, so what do all these equations actually mean and how are you supposed to understand them? Here is a fun video, using the secant line, which gives you an intuitive understanding of what the derivative actually is.

	Krista King Math: Secant line
video by Krista King Math

Before we discuss some details about the derivative, let's talk about notation.

Derivative Notation

You will see and use various types of notation to write this derivative. Most books just tell what to write without explaining why. Let me try to explain. In the following examples, I will assume \( y = f(x) \).

Various Notation Methods to Write the Derivative
\( f~'(x) \)	Used quite often and the small quote symbol indicates the derivative. Make sure that this symbol appears clearly in your work.
\( y' \)	Also used a lot. The disadvantage of this notation is that you need to pick up from the context what the variable of differentiation is, since it is not explictly stated in the expression, like it is in the previous example.
\( \displaystyle{\frac{d}{dx}[f(x)]} \)	Probably the clearest way to write the derivative but is also longer than the previous two ways. Use of this notation is usually limited to situations when it is important to clearly show the function itself and the variable of differentiation.
\( \displaystyle{\frac{dy}{dx}} \)	Also quite common. This is shorthand for \( \displaystyle{\frac{d[y]}{dx} }\) and \( \displaystyle{ \frac{d}{dx}[y]} \). [ see note below ]
\( D_x[y] \) and \( D_x[f(x)] \)	Not used much in beginning calculus. However, they are important when you study partial derivatives.
note - - The notation \( dy/dx \) does NOT mean dy divided by dx. It means the derivative of y with respect to x. Right before you begin studying integration, you will probably cover a section about differentials. This section will prove that you can write \(\displaystyle{\frac{dy}{dx} = f~'(x)} \) in differential form as \( dy = [ f~'(x) ] dx\). This can propagate the illusion of division but don't let yourself be fooled. It takes quite a bit to prove the differential form.

Using The Limit Definition

Okay, let's watch some videos related to the derivative and how to use the limit definition to get our heads around this concept.

This first video contains a great explanation of how to think about this limit definition and what it means. It also contains a good example of how to calculate and interpret the limit.

	Krista King Math: Derivatives
video by Krista King Math

After watching that first video, if the ideas are still not clear, here is another good video for you to watch explaining the same concepts.

	PatrickJMT: Understanding the Definition of the Derivative
video by PatrickJMT

Here is video that compares the graph of a function to the graph it's derivative. It is a great follow-up to the previous video.

	PatrickJMT: Sketching the Derivative of a Function
video by PatrickJMT

Fortunately, we are not going to have to use this limit definition to calculate derivatives all the time. There are some basic rules, based on the limit definition, that will very quickly replace the use the limit definition to calculate derivatives. Let's look at a few here.

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 \(f(x)=c\)

\(\displaystyle{ f~'(x) = }\) \(\displaystyle{ \lim_{h \to 0}{\frac{f(x+h) - f(x)}{h}} = }\) \(\displaystyle{ \lim_{h \to 0}{\frac{c - c}{h}} = \lim_{h \to 0}{0} = 0 }\)

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.

\(\displaystyle{ \frac{d}{dx}[cf(x)] = c \frac{d}{dx}[f(x)] }\)

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.

Addition and Subtraction Rules

When you have two functions that are added or subtracted, you just take the derivative of each individually. Mathematically, it looks like this.

\(\displaystyle{ \frac{d}{dx}[f(x) \pm g(x)] = }\) \(\displaystyle{ \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)] }\)

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. ]

Coming Up

Here is a great video with quick explanations of the next 4 or 5 rules that you will learn to calculate derivatives. This video is especially helpful if you are refreshing your calculus skills or just need a review before an exam. If you are first learning derivatives, it is better to skip this video for now and come back to it right before your first exam on derivatives.

	Krista King Math: Solving Derivatives
video by Krista King Math

After working some practice problems, your next logical step is the power rule.

next: power rule →

Practice Problems

Instructions - - Unless otherwise instructed, use the limit definition and the other rules on this page to calculate the derivative in these problems. Give your answers in exact terms, completely factored.