Derivative of Implicit functions

Implicit Functions:

  • Implicit functions are equations where the relationship between variables is not explicitly stated. Instead, it's presented as an equation involving both variables.
  • Typically represented as F(x,y)=0 or G(x,y)=0, where F or G could be any expression involving both x and y.

Derivative of Implicit Functions:

  • Finding the derivative of an implicitly defined function involves using implicit differentiation, treating y as a function of x even when y isn't explicitly written as a function of x.

Steps to Find dydx Implicitly:

  1. Differentiate both sides of the equation with respect to x:
    • Apply the chain rule whenever y appears.
  2. Solve for dydx:
    • Collect terms involving dydx on one side of the equation.

Example:

Given the equation x2+y2=25:

  1. Differentiate both sides with respect to x: ddx(x2+y2)=ddx(25)   2x+2ydydx=0

  2. Solve for dydx: dydx=xy

Tips:

  • Pay attention to chain rule usage:
    • Every term involving y must be multiplied by dydx when differentiating with respect to x.
  • Isolate the derivative:
    • After implicit differentiation, isolate dydx if possible to express it explicitly.
  • Watch for implicit relationships:
    • Implicit derivatives might reveal relationships between x and y not easily seen in explicit functions.

Special Cases:

  • Higher Derivatives:

    • After finding the first derivative dydx, further derivatives can be found by differentiating the equation again and solving for d2ydx2 or higher derivatives if needed.
  • Parametric Forms:

    • Sometimes it's helpful to express an implicitly defined function in parametric form, where x and y are defined separately as functions of a parameter.