Chain Rule Example


In the last section, the Chain Rule is introduced. It can calculate the derivative of a function when it is can be expressed in terms of another expression, such as y = (x +1)2 sin(x + 1).



Implicit Differentiation Example
  Suppose y is defined by a relation with x and it is hard to express the function y in terms of x. For example, x3 + y3 - 4xy = 0. In order to draw the diagram of function x3 + y3 - 4xy = 0, the diagram of function z = x3 + y3 - 4xy is plotted. The function z intersects the x-y plane at z = 0. The function expression and diagram shows that y is hard to be rewritten in term of x. In these cases, the derivative of y can be calculated by with the Implicit Differentiation method.

Function z = x3 + y3 - 4xy

Function x3 + y3 - 4xy = 0
    Implicit Differentiation

Implicit Differentiation

The implicit differentiation method states:

An equation f(x,y) = 0 defines y implicitly as a function of x. In order to to find the derivative of y, differentiate both sides of the original equation f(x, y) = 0 and solve the resulting equation for dy/dx. This differentiation method is known as implicit differentiation.

For example, given x3 + y3 - 4xy = 0, find dy/dx.

Differentiate both side of x3 + y3 - 4xy = 0 with respect to x, gives


Since y is implicitly defined by x, d(y3)/dx is not 0. Consider z = y3 and apply Chain Rule,


Recall that if F(x) = f(x)g(x), then F '(x) = f '(x)g(x) + f (x)g '(x).The formula can be used to calculate d(4xy)/dx. Consider f(x) = x and g(x) = y, thus


Substitute equation(2) and (3) into (1),



     3x2 + 3y2dy/dx - (4y + 4xdy/dx) = 0

     (3y2 - 4x)dy/dx = 4y - 3x2

     dy/dx = (4y - 3x2)/(3y2 - 4x)