Implicit Derivative

Implicit Derivative Assume that the equation f (x, y) = 0 represents y as an implicit function of x. If y is a differentiable function of x, then the equation f (x, y) = 0 is differentiated with respect to x and the value of dy / dx is obtained by solving this equation. Examples of implicit derivatives: - Question 1: - If x ^3 + 3 x ^2 y 2 y ^3 = 5, find dy / dx. Solution: - x ^3 + 3 x ^2 y 2 y ^3 = 5 (1) Differentiating both sides of (1) with respect to x. Or, d / dx (x ^3) +3 d /dx (x^2 y) -2 d/dx (y^3) = d/dx (5) Or, 3x^2 +3 {x^2 dy/dx + y d/dx(x^2)} 2 *3y^2 dy/dx =0 Or, 3 x^2+3 (x^2 dy/dx + y * 2x) 6 y^2 dy/dx = 0 Or, 3 x^2+3 x^2 dy/dx +6 x y- 6 y^2 dy/dx=0 Or, -3 dy/dx (2 y^2-x^2) + 3 x (x + 2 y)= 0 Or, -3 dy/dx (2 y^2-x^2) = -3 x (x + 2 y) Or, dy/dx = 3 x (x + 2 y)/ 3 (2 y^2-x^2) Therefore, dy/dx = x (x + 2 y)/ (2 y^2-x^2) Question 2: - If x + y = 1, find dy/dx. Solution: - Differentiating both sides with respect to x. d / dx (x) + d /dx (y) = d/dx (1) or, 1 + dy/dx = 0 or, dy/dx = -1 Therefore dy / dx = -1