Derivative of Parametric functions

Parametric Equations:

  • Parametric equations represent a relationship between two variables, often denoted as x and y, where each variable is expressed in terms of a third variable, usually t, known as the parameter.
  • Common parametric equations take the form: x=f(t) y=g(t)

Derivative of Parametric Functions:

  • To find the derivative of a parametric curve given by x=f(t) and y=g(t), the derivative dydx is calculated using the chain rule.
  • The derivative dydx represents the slope of the curve at any given point.

Steps to Find dydx:

  1. Derive both x and y with respect to t: dxdt=f(t) dydt=g(t)

  2. Calculate dydx using the chain rule: dydx=dydtdxdt=g(t)f(t)

Example:

Given parametric equations x=cos(t) and y=sin(t):

  1. Find dydx:

    dxdt=sin(t)
    dydt=cos(t)

  2. Calculate dydx:

    dydx=dydtdxdt=cos(t)sin(t)=tan(t)

Special Cases:

  • Vertical Tangents:
    • When dxdt=0 but dydt0, the curve has a vertical tangent.
  • Horizontal Tangents:
    • When dydt=0 but dxdt0, the curve has a horizontal tangent.

Tips:

  • Parametric Differentiation Rule:

    • Sometimes it's convenient to express y explicitly as a function of x and differentiate using implicit differentiation.
  • Substitute for t or Express in terms of x or y:

    • Substituting t with expressions involving x or y can help simplify calculations or analyze the curve's behavior at specific points.