http://www.ck12.org Chapter 2. Derivatives
Use implicit differentiation to findd^2 y/dx^2 if 5x^2 − 4 y^2 = 9 .Also find ddx^2 y 2
∣∣
∣(x,y)=( 2 , 3 ).What does the second
derivative represent?
Solution:
d
dx[^5 x(^2) − 4 y (^2) ] = d
dx[^9 ]
10 x− 8 ydydx= 0.
Solving fordy/dx,
dy
dx=
5 x
4 y.
Differentiating both sides implicitly again (and using the quotient rule),
d^2 y
dx^2 =
( 4 y)( 5 )−( 5 x)( 4 dy/dx)
( 4 y)^2
= 1620 yy 2 − 1620 yx 2 dydx
= 45 y− 45 yx 2 dydx.
But sincedy/dx= 5 x/ 4 y, we substitute it into the second derivative:
d^2 y
dx^2 =
5
4 y−5 x
4 y^2.5 x
4 y
d^2 y
dx^2 =5
4 y−25 x^2
16 y^3.This is the second derivative ofy.
The next step is to find:ddx^2 y 2
∣∣
∣(x,y)=( 2 , 3 )d^2 y
dx^2∣∣
∣∣
∣( 2 , 3 )=
5
4 ( 3 )−
25 ( 2 )^2
16 ( 3 )^3
= 275.
Since the first derivative of a function represents the rate of change of the functiony=f(x)with respect tox, the
second derivative represents the rate of change of the rate of change of the function. For example, in kinematics (the
study of motion), the speed of an object(y′)signifies the change of position with respect to time but acceleration
(y′′)signifies the rate of change of the speed with respect to time.