Implicit Differentiation

1
Implicit Differentiation

• The functions we considered so far can be
  described by expressing one variable explicitly
  in terms of another variable, e.g.
• y x2 3x or y cos x
• However, some functions are defined implicitly by
  a relation between x and y such as
• x2 y2 1 or 2y x2 sin y
• For most of this kind of functions, its not easy
  to solve for y to express it explicitly as a
  function of x.
• But fortunately we dont need to solve an
  equation for y in order to find the derivative of
  y. Instead we can use the method of implicit
  differentiation

2
Implicit Differentiation
y is expressed explicitly as a function of x.
y is expressed implicitly as a function of x.
To differentiate the implicit equation, we write
f (x) in place of y to get
3
Implicit Differentiation (cont.)
Now differentiate using the chain rule
which can be written in the form
subbing in y
Solve for y
4
Example 1
This is not a function, but it would still be
nice to be able to find the slope.
Do the same thing to both sides.
Note use of chain rule.
5
Example 2
This cant be solved for y.