8. Differentiation of Implicit Functions

by M. Bourne


We meet many equations where y is not expressed explicitly in terms of x only, such as:

math formula

You can see several examples of such expressions in the Polar Graphs section.

It is usually difficult, if not impossible, to solve for y so that we can then find math formula.

We need to be able to find derivatives of such expressions to find the rate of change of y as x changes. To do this, we need to know implicit differentiation.

 

Example 1:

Find the expression for math formula if y4 + x5- 7x2 - 5x-1 = 0.


Answer


Example 2:

Find the slope of the tangent at the point (2,-1) for the curve:

math formula


Answer


In LiveMath, we can see the shape of the curve we have been dealing with. We can also see that the slope of the tangent is indeed -4.

LIVEMath

 

Example 3 (Involves Product Rule)

Find the expression for math formula if:

math formula

(This is the example given above.)


Answer




Find Scholarships Today!


Book mark this page in Del.icio.us, Furl, Digg, StumbleUpon, whatever...


Didn't find what you are looking for? Try search:


Need a break? Play a math game. Well, they all involve math... No, really!

dumbolf memoTST bola shadow factory mindfields trick-hoops-challenge crystal clear