11DIFFERENTIATION - Department of Mathematics

EXAMPLE 5
SOLUTION ✔
11.6 IMPLICIT DIFFERENTIATION AND RELATED RATES 763
Substituting x 1 and y 2 into this equation gives
3(1) 2 2 dy
(2)
dx 2(1)(2)3 12(1) dy
dx
12 dy
dy
16 12
dx dx
and, solving for dy
dx ,
dy
dx 28
11
Note that it is not necessary to find an explicit expression for dy/dx.
REMARK In Examples 3 and 4, you can verify that the points at which we
evaluated dy/dx actually lie on the curve in question by showing that the
coordinates of the points satisfy the given equations.
Find dy/dx given that x and y are related by the equation
x 2 y 2 x 2 5
Differentiating both sides of the given equation with respect to x, we obtain
d
dx (x2 y2 ) 1/2 d
dx (x2 ) d
dx (5)
1
2 (x2 y2 1/2 d
)
dx (x2 y2 ) 2x 0
1
2 (x2 y2 ) 1/22x 2y dy
2x 0
dx
R ELATED R ATES
2x 2y dy
dx 4x(x2 y 2 ) 1/2
2y dy
dx 4x(x2 y 2 ) 1/2 2x
[Writing x 2 y 2
(x 2 y 2 ) 1/2 ]
(Using the general
power rule on the
first term)
[Transposing 2x and
multiplying both sides
by 2(x2 y2 ) 1/2 ]
dy
dx 2x x2 y2 x
y
Implicit differentiation is a useful technique for solving a class of problems
known as related rates problems. For example, suppose x and y are each
functions of a third variable t. Here, x might denote the mortgage rate and y
the number of single-family homes sold at any time t. Further, suppose we
have an equation that gives the relationship between x and y (the number of