Question

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

Answer 1

`(dy)/(dx)=(2t^3-e^(t-t^2)t^2(1-2t))/(t^2-te^(t^2-t+1)(2t-1)+e^(t^2-t+1))`

Explanation:

When a functiion is given in parametric form such as `f(t)=(x(t),y(t))`, its dervative is given by `(dy)/(dx)=((dy)/(dt))/((dx)/(dt))`

Here we have `y(t)=t^2-e^(t-t^2)` hence `(dy)/(dt)=2t-e^(t-t^2)(1-2t)`

and `x(t)=t-(e^(t^2-t+1))/t` hence `(dx)/(dt)=1-(te^(t^2-t+1)(2t-1)-e^(t^2-t+1))/t^2`

= `(t^2-te^(t^2-t+1)(2t-1)+e^(t^2-t+1))/t^2`

Hence `(dy)/(dx)=(t^2(2t-e^(t-t^2)(1-2t)))/(t^2-te^(t^2-t+1)(2t-1)+e^(t^2-t+1))`

= `(2t^3-e^(t-t^2)t^2(1-2t))/(t^2-te^(t^2-t+1)(2t-1)+e^(t^2-t+1))`