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

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

Differentiation 113

TIP • Always write out all formulas in your solutions.

Power Rule

Iff(x)=cwherecis a constant, then f′(x)=0.

Iff(x)=xnwherenis a real number, thenf′(x)=nxn−^1.

Iff(x)=cxnwherecis a constant andnis a real number, then f′(x)=cnxn−^1.

Summary of Derivatives of Algebraic Functions

d

dx

(c)=0,

d

dx

(xn)=nxn−^1 , and

d

dx

(cxn)=cnxn−^1

Example 1

Iff(x)= 2 x^3 , find (a)f′(x), (b) f′(1), and (c) f′(0).

Note that (a)f′(x)= 6 x^2 , (b)f′(1)=6(1)^2 =6, and (c)f′(0)=0.

Example 2

Ify=

1

x^2

, find (a)

dy

dx

and (b)

dy

dx

|x= 0 (which represents

dy

dx

atx=0).

Note that (a)y=

1

x^2

=x−^2 and thus,

dy

dx

=− 2 x−^3 =

− 2

x^3

and (b)

dy

dx

|x= 0 does not exist because

the expression

− 2

0

is undefined.

Example 3

Here are several examples of algebraic functions and their derivatives:

DERIVATIVE WITH

FUNCTION WRITTEN INcxnFORM DERIVATIVE POSITIVE EXPONENTS

3 x 3 x^13 x^0 = 33

− 5 x^7 − 5 x^7 − 35 x^6 − 35 x^6

8

√

x 8 x

(^12)
4 x−
(^124)
x
12 or

4

√

x

1

x^2

x−^2 − 2 x−^3

− 2

x^3

− 2

√

x

− 2

x

12 =−^2 x

−^12 x−^321

x

32 or

1

√

x^3

44 x^000

π^2 (π^2 )x^000