A Note It may be noted here that dy dx is expressed in terms of parameter only without directly involving the main variables x and y . Example Find , if . Solution Let x = a cos θ , y = a sin θ . Then ( cos ( sin (cos (sin θ + θ = Hence, x = a cos θ , y = a sin θ is parametric equation of Now d θ = – a cos θ sin θ and dy d θ = a sin θ cos θ Therefore sin tan cos θ = = − θ = − EXERCISE .
If x and y are connected parametrically by the equations given in Exercises to , without eliminating the parameter, Find dy dx . . x = at , y = at . x = a cos θ , y = b cos θ .
x = sin t , y = cos t . x = t , y = . x = cos θ – cos θ , y = sin θ – sin θ . x = a ( θ – sin θ ), y = a ( + cos θ ) .
x = cos2 t , cos2 . log tan y = a sin t . x = a sec θ , y = b tan θ . x = a (cos θ + θ sin θ ), y = a (sin θ – θ cos θ ) .
If , , show that = − . Second Order Derivative Let y = f ( x ). Then dx = f ′ ( x ) ... ( ) If f ′ ( x ) is differentiable, we may differentiate ( ) again w.r.t.
x . Then, the left hand side becomes d which is called the second order derivative of y w.r.t. x and is denoted by . The second order derivative of f ( x ) is