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Consider a function

x_dot= f (x), its 1st derivative can be written as x

^{(1)}=f(x),And its 2nd derivative can be x

^{(2)}=f '(x). x_dot,And recursively, we can find out x(n) nth derivative of the x_dot= f(x) in the case if f(x) is linear, which is a reason for the formation of matrix exponential (e

^{AT}) If A is a linear matrix in f (x).Or

One can also say that if f (x) results in a closed form expression for its Taylor expansion. Then nth derivative can be written. My question is that expression can be written for nonlinear systems if they come to have a closed form expression in their Taylor expansion.