Cracking The Ap Calculus ab Exam 2018

In calculus, you’ll be asked to do two things: differentiate and integrate. In this section, you’re going to
learn differentiation. Integration will come later, in the second half of this book. Before we go about the
business of learning how to take derivatives, however, here’s a brief note about notation. Read this!

NOTATION

There are several different notations for derivatives in calculus. We’ll use two different types
interchangeably throughout this book, so get used to them now.

We’ll refer to functions three different ways: f(x), u or v, and y. For example, we might write: f(x) = x^3 ,

g(x) = x^4 , h(x) = x^5 . We’ll also use notation like: u = sin x and v = cos x. Or we might use: y = .

Usually, we pick the notation that causes the least confusion.

The derivatives of the functions will use notation that depends on the function, as shown in the following
table:

Function First  Derivative Second   Derivative

f(x) f′(x) f′′(x)

g(x) g′(x) g′′(x)

y y′    or   y′′    or  

In addition, if we refer to a derivative of a function in general (for example, ax^2 + bx + c), we might
enclose the expression in parentheses and use either of the following notations:

(ax^2   +   bx  +   c)′,    or   (ax^2  +   bx  +   c)

Sometimes math books refer to a derivative using either Dx or fx. We’re not going to use either of them.

THE POWER RULE

In the last chapter, you learned how to find a derivative using the definition of the derivative, a process
that is very time-consuming and sometimes involves a lot of complex algebra. Fortunately, there’s a
shortcut to taking derivatives, so you’ll never have to use the definition again—except when it’s a
question on an exam!

The basic technique for taking a derivative is called the Power Rule.