MATH 125 Lecture Notes - Lecture 22: Implicit Function, Distributive Property

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22 Feb 2017
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# section 3. 7: another example of implicit differentiation. # the quadrifolium curve is also known as the four-leaved clover. # it is the set of points for which the following polynomial is 0: quadrifolium := (x^2+y^2)^3 - (x^2-y^2)^2; quadrifolium := x. # here"s a computer-generated graph to explain the curve"s name: with( plots ): implicitplot( quadrifolium, x=-22, y=-22, numpoints=100000 ); # let"s find the derivative y" using implicit differentiation. # ( x^2 + y^2 )^3 - ( x^2 - y^2 )^2 = 0. # differentiate with respect to x using the chain rule: # 3 (x^2 + y^2)^2 (2x + 2yy") - 2 (x^2 - y^2) (2x - 2yy") = 0. # expand the square (the first factor of the first term): # 3(x^4 + 2x^2y^2 + y^4)(2x + 2yy") - 2(x^2 - y^2)(2x - 2yy") = 0. # - 2 ( 2x^3 - 2x^2yy" - 2xy^2 + 2y^3y" ) = 0.

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