# If x = f(t), y = g(t) are differentiable functions of parammeter ‘ t ’ then prove that y is a differentiable function of 'x' and  hence, find dy/dx if x=a cost, y=a sint - Mathematics and Statistics

If x = f(t), y = g(t) are differentiable functions of parammeter ‘ t ’ then prove that y is a differentiable function of 'x' and  hence, find dy/dx if x=a cost, y=a sint

#### Solution

Given x=f(t),y=g(t) are differentiable function of parameter 't'

x=acost and y=asint

find dy/dx=?

x=acost

differentiate x w.r.t 't'

dx/dt=d/dt(acost)

dx/dt=asint................(1)

y=asint

dy/dt=d/dt (asint)

dy/dt=-acost.............(2)

dividing equation 2 by 1

(dy/dt)/(dx/dt)=(-acost)/(asint)=-cost/sint......(3)

now " "x=acost

therefore cost=x/a

y=asint

therefore sintt=y/a

from equation 3

dy/dx=-(x/a)/(y/a)=-x/y

Concept: Derivatives of Functions in Parametric Forms
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