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

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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.
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