At this point, you should be able to recognize quite a few ways to write the derivative:
...and so on. However, on the AP exam, you need to know more than just how to write or take the derivative of a function. In this article, I'll be going over the definitions of the derivative and some common derivatives that you'll need to know for the AP exam.
Defining the Derivative
Both definitions of the derivative that you should know center around the idea of the derivative as the slope of the secant line passing between two points. To find the instantaneous rate of change (A.K.A, the derivative) for a spot on the graph between those two points, the points must grow infinitesimally closer together. Put another way, the space between them must approach 0 - effectively, the two points have now "become" the one point you are trying to find the derivative of.
The Limit Definition
This is the definition that most often appears on the multiple-choice and free-response sections of the AP exam, and you will either have to identify it out of other incorrect choices or use it to find a derivative. It is as follows:
Since the slope of the secant line between two points is the change in y divided by the change in x, the expression without the limit is simply the equation for the secant line: f(x) represents your "initial" y-value, and f(x+delta x) represents your "final" y-value.
- Once you add the limit to the expression, the formula's use as the definition of the derivative becomes clear: it is the slope of the secant line as the distance between the two points (delta x) approaches 0. Thus, this formula is used to approximate the instantaneous rate of change (the derivative) of a point on the graph.
- This definition may also be referred to as the difference quotient method
The Alternate Definition
The second definition of the derivative that you should be able to recognize is the alternate definition (I know, it's a really creative name). It looks like this:
The limit definition appears much more often on the AP exam than the alternate definition, but you should still be able to recognize and use both of them.
Calculating Basic Derivatives
With taking a derivative, there are a couple of rules that you'll need to know
Let's go over a quick example then:
Sum and Difference Rule
This rule is pretty straight forward and is used when you're taken a derivative of a function that has multiple terms.
The Product Rule
This can get pretty confusing through just words so we'll go over a quick example to help see how it should be done!
Just like the product rule, this one's a little bit of a doozy and so my teacher taught us a song to help remember this (it's meant to go with the Snow White and the Seven Dwarfs HeIgh-Ho song): "You take the low, d hi, minus the high, d low, square the bottom and away we go, heigh-ho, heigh-ho!"
And that's it! These were the basic rules for taking a derivative that you'll need to know in order to get that five! If you need some additional practice, check out the resources linked below!
What Should I Memorize?
The chart below has the most common derivatives that appear on the exam that you should know. Remember, these are listed in their most basic forms, so be prepared to see them with negative signs and coefficients in front of them and adjust your derivative accordingly. You should also know how to use these derivatives in conjunction with other functions (such as when using the product rule, quotient rule, or chain rule). Finally, you may notice that these functions seem quite familiar. This is because they are all either parent functions
or trig functions
to have memorized!
For the AP Calculus AB exam, it is not required to have these derivatives memorized, but it will certainly make solving problems where you need to take the derivative of a function a lot faster. Again, you should be able to take the derivatives of these functions with modifications (negative signs and coefficients) and using the product, quotient, or chain rules. If you're familiar with these common derivatives, rules, and definitions, your year in AP Calculus will be a breeze.