3.6 Implicit Differentiation

Explicit Function

An explicit function can be solved for y as a function of x …. y = f(x)

Implicit Function

Often the implicit function(s) are part of a relation. That is, there is more than one function defined for the relation … often with different domains. Text visualizes on page 185.

defines at least the two functions

Graphically:

Often it is difficult (if not impossible!!) to "solve for y".

To differentiate an implicit equation, leave as an equation, and differentiate implicitly.

BE CAREFUL

Examples:

BE CAREFUL OF PRODUCTS!!

#22 If

#34 Find the second derivative by implicit differentiation.

Find the slope of the tangent line to the graph of the equation at P.

Note: If the problem requests only the equation of the line at a given point, it is not necessary to find the general form of dy/dx in terms of x. Instead, after the first step of implicit differentiation, substitute the value of the given point into x and dy to find the slope at the GIVEN point and write the equation of the line.

at P(-1,3) at P(1,0)

Assignment: 3.6 pg 169; 1, 5, 7, 11-15 odd, 19, 21, 25, 27, 33, 35

**Work to a high comfort level with these, because the next section APPLIES them in Related Rate problems!!