How do you differentiate the following parametric equation: # x(t)=e^t/(t+t)^2-t, y(t)=t-e^(t) #?
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#dx/dt = (e^t)/(4t^2) - (e^t)/(2t^3) - 1#, #dy/dt = 1 - e^t#
Because the curve is expressed in terms of two functions of #t# we can find the answer by differentiating each function individually with respect to #t#. First note that the equation for #x(t)# can be simplified to:
#x(t) = 1/4 e^t 1/(t^2) - t#
While #y(t)# can be left as:
#y(t) = t - e^t#
Looking at #x(t)#, it is easy to see that the application of the product rule will yield a quick answer. While #y(t)# is simply standard differentiation of each term. We also use the fact that #d/dx e^x = e^x#.
#dx/dt = (e^t)/(4t^2) - (e^t)/(2t^3) - 1#
#dy/dt = 1 - e^t#
