Consider the parametric curve given by x = t - e^(t), y = 7t + 7e^(-t) Find dy/d

Question

Consider the parametric curve given by x = t - e^(t), y = 7t + 7e^(-t)

Find dy/dx and d^(2)y/dx^(2) in terms of t.

dy/dx= (7(1 - e^(-t))/(1 - e^(t))

d^(2)y/dx^(2)=?????

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Explanation / Answer

x = t - e^(t), y = 7t + 7e^(-t) dx/dt = 1 - e^t ==>dt/dx = 1/[1-e^t] dy/dt = 7 - 7e^-t = 7[1 - e^-t] dy/dx = dy/dt/dx/dt = 7[1 - e^-t]/[1 - e^t] d^y/dx^2 = 7[0 + e^-t*dt/dx]*1/[1 - e^t] + 7[1 - e^-t]*-[0-e^t*dt/dx]/[1 - e^t]^2 = 7[e^-t * dt/dx]*1/[1 - e^t] + 7[1 - e^-t]*[e^t*dt/dx]/[1 - e^t]^2 = 7[e^-t * 1/[1-e^t]]*1/[1 - e^t] + 7[1 - e^-t]*[e^t*1/[1-e^t]]/[1 - e^t]^2 = 7e^-t/[1 - e^t]^2 + 7[1 - e^-t]*e^t/[1 - e^t]^3 = [ 7 e^-t - 7 + 7e^t - 7]/[1 - e^t]^3 = [7 e^-t + 7e^t - 14]/[1- e^t]^3

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