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





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