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 x^{3} + y^{3 }- 4xy
= 0, find dy/dx.
Differentiate both side of x^{3} + y^{3 }- 4xy = 0 with
respect to x, gives
(1)
Since y is implicitly defined by x, d(y^{3})/dx is not 0. Consider
z = y^{3} and apply Chain Rule,
(2)
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
(3)
Substitute equation(2) and (3) into (1),
so
3x^{2} +
3y^{2}dy/dx
- (4y + 4xdy/dx) = 0
(3y^{2 }- 4x)dy/dx = 4y - 3x^{2}
dy/dx = (4y - 3x^{2})/(3y^{2 }-
4x) |