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

Question

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

Ezra Adams · Accepted Answer

Hi there!

Anytime you're given parametric equations to differentiate, you use the relationship whereby:

#dy/(dx) = (dy/(dt))/(dx/(dt)# Let's start off by differentiating each equation separately, then combine the two afterwards! Differentiating y with respect to t (dy/dt) we get: # dy/dt = -2t # Differentiating x with respect to t (dx/dt) we get: #dx/dt = e^t - 1/t^2 # (If you're not sure how I got the # -1/t^2 # from #1/t#... If you convert the #1/t# to #t^-1# and differentiate using the power rule, it comes out to #-t^-2# which is equivalent to #-1/t^2#) Now putting everything together we get: #dy/dx = (-2t)/(e^t - 1/t^2) # Hopefully this answers your question clearly! If you have any follow up questions, please feel free to ask! :)

Luke Alvarez · Answer

undefined To differentiate the parametric equations x(t) = e^t - 1/t and y(t) = 1 - t^2 with respect to t, you differentiate each equation separately using the chain rule for x(t) and the power rule for y(t).