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:
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 
.
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 if y4 + x5-
7x2 -
5x-1 = 0.
Example 2:
Find the slope of the tangent at the point (2,-1) for the curve:
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.
Example 3 (Involves Product Rule)
Find the expression for
if:
(This is the example given above.)
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