76 STEP 4. Review the Knowledge You Need to Score High
6.1 Derivatives of Algebraic Functions
Main Concepts:Definition of the Derivative of a Function; Power Rule; The Sum,
Difference, Product, and Quotient Rules; The Chain Rule
Definition of the Derivative of a Function
The derivative of a functionf, written asf′, is defined as
f′(x)=limh→ 0
f(x+h)−f(x)
h
,
if this limit exists. (Note thatf′(x) is read asf prime ofx.)
Other symbols of the derivative of a function are:
Dxf,
d
dx
f(x), and if y=f(x),y′,
dy
dx
, andDxy.
Letmtangentbe the slope of the tangent to a curvey= f(x) at a point on the curve. Then,
mtangent=f′(x)=hlim→ 0
f(x+h)−f(x)
h
mtangent(atx=a)=f′(a)=limh→ 0
f(a+h)−f(a)
h
or limx→a
f(x)−f(a)
x−a
.
(See Figure 6.1-1.)
y f(x)
x
tangent
(a, f(a))
Slope of tangent to f(x)
at x = a is m = f ' (a)
0
Figure 6.1-1
Given a functionf,iff′(x) exists atx=a, then the functionfis said to be differentiable at
x=a. If a functionfis differentiable atx=a, thenfis continuous atx=a. (Note that the
converse of the statement is not necessarily true, i.e., if a functionf is continuous atx=a,
thenf may or may not be differentiable atx=a.) Here are several examples of functions
that are not differentiable at a given numberx=a. (See Figures 6.1-2–6.1-5 on page 77.)