11DIFFERENTIATION - Department of Mathematics

S ELF-CHECK E XERCISES 11.6
11.6 Exercises
11.6 IMPLICIT DIFFERENTIATION AND RELATED RATES 767
Step 4 Next, we differentiate the equation x2 y2 40002 with respect to
t, obtaining
2x dx dy
2y
dt dt
(Remember, both x and y are functions of t.)
Step 5 Substituting x 5000, y 3000, and dy/dt 600, we find
2(5000) dx
2(3000)(600)
dt
dx
360
dt
Therefore, the distance between the rocket and the spectator is
changing at a rate of 360 feet/second.
Be sure that you do not replace the variables in the equation found in step
3 by their numerical values before differentiating the equation.
1. Given the equation x3 3xy y3 4, find dy/dx by implicit differentiation.
2. Find an equation of the tangent line to the graph of 16x2 9y2 144 at the point
2, 45
3 .
In Exercises 1–8, find the derivative dy/dx
(a) by solving each of the given implicit equations
for y explicitly in terms of x and (b) by
differentiating each of the given equations
implicitly. Show that, in each case, the results
are equivalent.
1. x 2y 5 2. 3x 4y 6
3. xy 1 4. xy y 1 0
5. x3 x2 xy 4 6. x2y x2 y 1 0
7. x
y x2 1 8. y
x 2x3 4
In Exercises 9–30, find dy/dx by implicit differentiation.
9. x2 y2 16 10. 2x2 y2 16
11. x2 2y2 16 12. x3 y3 y 4 0
Solutions to Self-CheckExercises 11.6 can be found on page 770.
13. x 2 2xy 6 14. x 2 5xy y 2 10
15. x 2 y 2 xy 8 16. x 2 y 3 2xy 2 5
17. x 1/2 y 1/2 1 18. x 1/3 y 1/3 1
19. x y x 20. (2x 3y) 1/3 x 2
21. 1 1
1 1
1 22. 5
2 2 3 3 x y x y
23. xy x y 24. xy 2x y 2
25.
x y
3x 26.
x y
x y
2x
2x 3y
27. xy 3/2 x 2 y 2 28. x 2 y 1/2 x 2y 3
29. (x y) 3 x 3 y 3 0
30. (x y 2 ) 10 x 2 25