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August 4, 2020
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.
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.
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.
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:
This definition more clearly shows the equation of the secant line (change in y over change in x). By taking the limit as x approaches c, the two points are getting infinitesimally closer together and effectively "become" one point whose derivative you are approximating.
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.
With taking a derivative, there are a couple of rules that you'll need to know
The formula for the power rule looks something like this
With this formula, 'x' is the variable, and 'n' is the exponent
Let's go over a quick example then:
We can see that the we have an exponent of two and a coefficient of 3.
Last but not least, we have to subtract one from the exponent value
And voila! You've taken a derivative using the power rule!
This rule is pretty straight forward and is used when you're taken a derivative of a function that has multiple terms.
The rule in its formal form looks like this
The first one shows the sum rule while the second one shows the difference rule (That apostrophe means 'the derivative of')
This rule is used when two parts of a function are being multiplied by each other (this is different from chain rule!)
This looks super confusing but the saying my teacher taught me really helped: "first times the derivative of the second plus second times the derivative of the first"
This can get pretty confusing through just words so we'll go over a quick example to help see how it should be done!
This first step is to identify our terms. We have 'x^2' and 'e^x' and we can see that they are being multiplied by each other. Now, we'll assume that 'x^2' is going to be our 'first' and 'e^x' is going to be 'second'
Now, we have to follow through with the formula and saying: "first times the derivative of the second plus second times the derivative of the first"
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!"
This rule is more about identifying when you should use it and following the formula.
Last but not least, we have chain rule. This is definitely the one that most people have the hardest time with so it's important that you can get the basics of it first.
This would sound something like "the derivative of f of g of x equals to the f prime of g of x times g prime of x"
First, we need to identify our inner and outer functions. If we treat what's in the inner parentheses as one function and replace the whole with the variable 'x' for now.
We know how to take the derivative of this (power rule!) so we have the first part of this problem done
The last step is to multiply these two parts together
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!
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.
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