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

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

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

TIP • You do not have to answer every question correctly to get a 5 on the AP Calculus AB

exam. But always select an answer to a multiple-choice question. There is no penalty

for incorrect answers.

7.6 Higher Order Derivatives

If the derivative f′of a function f is differentiable, then the derivative of f′is the sec-

ond derivative of f represented by f′′(reads as f double prime). You can continue to

differentiatef as long as there is differentiability.

Some of the Symbols of Higher Order Derivatives

f′(x),f′′(x),f′′′(x),f(4)(x)

dy

dx

,

d^2 y

dx^2

,

d^3 y

dx^3

,

d^4 y

dx^4

y′,y′′,y′′′,y(4)

Dx(y),D^2 x(y),D^3 x(y),D^4 x(y)

Note that

d^2 y

dx^2

=

d

dx

(

dy

dx

)

or

dy′

dx

.

Example 1

Ify= 5 x^3 + 7 x−10, find the first four derivatives.

dy

dx

= 15 x^2 +7;

d^2 y

dx^2

= 30 x;

d^3 y

dx^3

=30;

d^4 y

dx^4

= 0

Example 2

Iff(x)=

√

x, findf′′(4).

Rewrite:f(x)=

√

x=x^1 /^2 and differentiate:f′(x)=

1

2

x−^1 /^2.

Differentiate again:

f′′(x)=−

1

4

x−^3 /^2 =

− 1

4 x^3 /^2

=

− 1

4

√

x^3

and f′′(4)=

− 1

4

√

43

=−

1

32

.

Example 3

Ify=xcosx, findy′′.

Using the product rule,y′=(1)(cosx)+(x)(−sinx)=cosx−xsinx

y′′=−sinx−[(1)(sinx)+(x)(cosx)]

=−sinx−sinx−xcosx

=−2 sinx−xcosx.

Or, you can use a calculator and enterd[x∗cosx,x, 2] and obtain the same result.