Download 1 − = − yx 1 = xy 1 x y + 1 x y +

Implicit Differentiation
A function f whose values y depend on the variable x according to an equation in terms of x and y is said to
be implicitly defined by the equation in question.
Sometimes, but not always, it is possible to solve y in terms of x from the given equation. Then one gets an
explicit expression for the function in question.
x 2 − y = −1 Explicit form: y = x 2 + 1
x2 + y 2 =
1does not have an explicit form since x 2 + y 2 =
1 defines two functions:
Implicit form:
But
y=
± 1 − x2
How can we differentiate implicit functions like
x2 + y 2 =
1?
Guidelines for implicit differentiation:
1.
Differentiate both sides with respect to x.
2.
Collect all dy/dx on the left side and other terms on the right.
3.
Factor dy/dx out of the left side.
4.
Solve for dy/dx.
Can the following represent y as a differentiable function of x: (remember functions are not differentiable at
undefined slopes - vertical lines or where it is not continuous.)
a)
x2 + y 2 =
0
b)
x2 + y 2 =
1
Example 1: Find the slope of the tangent line for
(x
2
c)
x + y2 =
1
+ y 2 ) + 3x 2 y − y 3 =
0 at ( 0,1)
2
Practice:
Use implicit differentiation to find the tangent line to the
(
)
2
(
lemniscate 2 x 2 + y 2 = 25 x 2 − y 2
)
at the point
(3,1) .
Example 2:
Determine if sin y = x is differentiable, if so find the largest interval of the form
where y is a differentiable function of x.
Second Derivative Implicitly:
d2y
Given x + y =
25 , find 2 .
dx
2
2
−a < y < a