Question

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

Answer 1

`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))`