CHAPTER 12
Higher-Order Derivatives
and Implicit Differentiation
12.2
12.3
12.4
12.5
The general pattern is
75
12.1
Find the second derivative y" of the function
by direct computation.
quotient rule,
By the chain rule,
By the
Use implicit differentiation to solve Problem 12.1.
y
2 = x
2 + 1. Take the derivative of both sides with respect to x. By the chain rule,
Thus, 2yy' = 2x, and, therefore, yy' =x. Take the derivative with respect to x of both sides, using the
product rule on the left: yy" + y'-y' — \. So, yy" = 1 - (y')
2 . But, since yy'— x, y' = xly. Hence,
yy" = 1 - x
2 /y
2 = (y
2 - x
2 )/y
2 = 1 ly
2 = 1 /(x
2 + 1). Thus, y" = 1 ly(x
2 + 1) = 1 l(x
2 + I)
3 '
2 .
Find all derivatives y'"' of the function y = irx
3 — Ix.
Find all derivatives y
(n) of the function
y' = 3irx
2 -7, y" = 6trx, y'" = 6ir, and y
(n) =0 for n>4.
This is enough to detect the general pattern:
and
Find all derivatives y
y = (3 + *r'
The general pattern is
12.6
Find all derivatives y
( "'of the function y = (x + l)/(x - 1).