Discuss Implicit Differentiation.

Discuss Implicit Differentiation.

In calculus, a method called implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.  y can be given as a function of x implicitly rather than explicitly. When we have an equation R(x, y) = 0, we may be able to solve it for y and then differentiate. However, sometimes it is simpler to differentiate R(x, y) with respect to x and y and then solve for dy/dx.
Example: Assume that y is a function of x . Find y' = dy/dx for x³ + y³ = 4 .
SOLUTION 1 : Begin with x3 + y3 = 4 . Differentiate both sides of the equation, getting

D ( x3 + y3 ) = D ( 4 ) ,

D ( x3 ) + D ( y3 ) = D ( 4 ) ,

(using the chain rule on D ( y3 ) .)

3x2 + 3y2 y' = 0 ,

so that (Now solve for y' .)

3y2 y' = - 3x2 , and

y' = -3x²/3y² = -x²/y² Ans.