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

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Answer 1 · 2016-05-07T11:10:04

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

When equation are given in parametric form such as

#x(t)=tsqrt(t^2-1)# and #y(t)=sqrt(t^2-e^t)#

Then #(dy)/(dx)=((dy)/(dt))/((dx)/(dt))#

#(dy)/(dt)=1/2xx1/sqrt(t^2-e^t)xx(2t-e^t)#

#(dx)/(dt)=1xxsqrt(t^2-1)+txx1/2xx1/sqrt(t^2-1)#

= #sqrt(t^2-1)+t/2xx1/sqrt(t^2-1)#

Hence #(dy)/(dx)=(1/(2sqrt(t^2-e^t))xx(2t-e^t))/(sqrt(t^2-1)+t/2xx1/sqrt(t^2-1))#

= #((2t-e^t)sqrt(t^2-1))/((t^2-1+t)sqrt(t^2-e^t))#