Continuity and Differentiability

Derivatives of Functions in Parametric Forms

Sometimes the relation between two variables is neither explicit nor implicit, but some link of a third variable with each of the two variables, separately, establishes a relation between the first two variables. In such a situation, we say that the relation between them is expressed via a third variable. The third variable is called the parameter. More precisely, a relation expressed between two variables x and y in the form x = f(t), y = g(t) is said to be parametric form with t as a parameter.
In order to find derivative of function in such form, we have by chain rule.
      \frac{dy}{dt}= \frac{dy}{dx}\frac{dx}{dt}

 \frac{dy}{dx}= \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\tt \left(whenever\ \frac{dx}{dt}\neq 0\right)

Thus \frac{dy}{dx}=\frac{g'(t)}{f'(t)}\ \left(as\ \frac{dy}{dt}=g'(t)\ and\ \frac{dx}{dt}=f'(t)\right) [provided f'(t) ≠ 0]

Part1: View the Topic in this video From 49:18 To 53:58

Part2: View the Topic in this video From 00:40 To 05:53

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