5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 9, 2018 10:9

110 STEP 4. Review the Knowledge You Need to Score High

7.1 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)=limh→ 0

f(x+h)−f(x)

h

mtangent(atx=a)= f′(a)=lim

h→ 0

f(a+h)−f(a)

h

or limx→a

f(x)−f(a)

x−a

.

(See Figure 7.1-1.)

y f(x)

x

tangent

(a, f(a))

Slope of tangent to f(x)

at x = a is m = f ′(a)

0

Figure 7.1-1

Given a functionf,iff′(x) exists atx=a, then the function fis said to be differentiable

atx=a. If a functionf is differentiable atx=a, thenf is continuous atx=a. (Note

that the converse of the statement is not necessarily true, i.e., if a functionf is continuous

atx=a, then f 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 7.1-2---7.1-5.)