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. dt = dy dx dx dt or whenever ≠ Thus as ( ) and g t g t f t f t ′ ′ ′ ′ [provided f ′ ( t ) ≠ ] Example Find dy dx , if x = a cos θ , y = a sin θ . Solution Given that x = a cos θ , y = a sin θ Therefore d θ = – a sin θ , dy d θ = a cos θ Hence cot θ = = − Example Find dy , if x = at , y = at . Solution Given that x = at , y = at So dt = at and dy dt = a Therefore at Example Find dy , if x = a ( θ + sin θ ), y = a ( – cos θ ).
Solution We have dx d θ = a ( + cos θ ), dy d θ = a (sin θ ) Therefore tan ( cos ) θ =