94 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 BC
exam. But always select an answer to a multiple-choice question. There is no penalty
for incorrect answers.6.6 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^3and 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.