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Its called " implicit " because we usually do not have a "nice" EXplicit form of y=
a good example is a circle:
x^2 +y^2 =25
To get an explicit form of y = we would have to break it into two functions, neither of which is pretty, and find y' with various ? forms ... ugh!
In Implicit differentiation, we know a relation that implies this circle : x^2 +y^2 =25
and there are 4 steps :
a) Differentiate BOTh sides wrt x, using chain rule, product rule, etc
b) collect all the dy/dx terms on one side, all other terms on other
c) factor out the dy/dx, and
d) Solve for dy/dx ( usually by dividing by the factor in c)
I usually tell MY students to include the dx/dx term even though it becomes 1, in order to make the process clear.
Example : What is the slope of the circle ( above ) at (3,-4) ?
Solution :
D(LS) = d(RS )
so : d/dx ( x^2 +y^2) = d(25)/dx
2x(dx/dx) +2y(dy/dy) =0 ( step a)
2y(dy/dx) =-2x(dx/dx ( step b
= -2x
and dy/dx = -2x/2y ( step c, d )
and now we substitute in ( x,y) = ( 3,-4)
so dy/dx = -2x/2y = -2(3)/2(4) = -3/4
Advantages of Implicit : faster, simpler, no complex forms, can be used if there is NO explicit form , and a " neater " answer
Disadvantages of implicit : Need to know product rule, chain rule, algebra collecting, etc, and to get an exact numerical answer you need to know BOTH an x and a y ON the curve . :
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