You can do it!!!

1
2.5 Implicit Differentiation
You can do it!!!
2
How would you find the derivative in the
equation x2 2y3 4y 2 where it is very
difficult to express y as a function of x?
To do this, we use a procedure called
implicit differentiation.
This means that when we differentiate
terms involving x alone, we can differentiate as
usual. But when we differentiate terms involving
y, we must apply the Chain Rule.
Watch the examples very carefully!!!
3
Differentiate the following with respect to x.
3x2 2y3 x 3y xy2
6x
6y2
y
1 3y
x(2y)y y2(1)
2xyy y2
Product rule
4
Find dy/dx given that y3 y2 5y x2 -4
Isolate dy/dxs
Factor out dy/dx
5
What are the slopes at the following points?
(2,0) (1,-3) x 0 (1,1)
undefined
6
Determine the slope of the tangent line to the
graph of x2 4y2 4 at the point
.
-2 -1 1 2
7
Differentiate sin y x
Product Rule
Differentiate x sin y y cos x
x cos y (y) sin y (1) y (-sin x) cos x
(1)(y)
x cos y (y) - cos x (y) -sin y - y sin x
y(x cos y - cos x) -sin y - y sin x
8
Given x2 y2 25, find y
Now replace y with
Multiply top and bottom by y
What can we substitute in for x2 y2?