Question

Find dy/dx and d2y/dx2. x = et, y = te−t

Answer 1

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Below is the solution:

x = e^t 
y = te^-t 

dy/dx = y'/x' 
= (1-t)e^-t/e^t 
= (1-t)e^-2t 

d^2/dx^2 = (x'y"-x"y')/x'^3 
= ((e^t)(t-2)e^-t - (e^t)(1-t)e^-t)/e^3t 
= (2t-3)e^-3t 

or, do it directly 

x = e^t, so 
y = lnx/x 

y' = (1-lnx)/x^2 = (1-t)e^-2t 
y" = (2lnx-3)/x^3 = (2t-3)e^-3t