Implicit Derivative

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 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



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