How do you differentiate the following parametric equation: # x(t)=t/sqrt(t^2-1), y(t)= sqrt(t^2-e^(t) #?

1 Answer
Oct 22, 2017

#dy/dx=((e^t-2t)(sqrt(t^2-1)^3))/(2sqrt(t^2-e^t))#

Explanation:

I assume that you want to find #dy/dx#.

Given the parametric equations #x(t)# and #y(t)#, #dy/dx=dy/dt*dt/dx=(dy/dt)/(dx/dt)#.

We are given #y(t)=sqrt(t^2-e^t)#. Using the chain rule, #dy/dt=(2t-e^t)/(2sqrt(t^2-e^t))#.

Using #x(t)=t/sqrt(t^2-1)# and the quotient rule, #dx/dt=(sqrt(t^2-1)-t^2/sqrt(t^2-1))/(t^2-1)=-1/sqrt(t^2-1)^3#.

Thus, #dy/dx=(dy/dt)/(dx/dt)=((2t-e^t)/(2sqrt(t^2-e^t)))/(-1/sqrt(t^2-1)^3)=(2t-e^t)/(2sqrt(t^2-e^t))*-sqrt(t^2-1)^3=((e^t-2t)(sqrt(t^2-1)^3))/(2sqrt(t^2-e^t))#