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)#?

1 Answer
Nov 24, 2017

#(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))#