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We know the derivative of e is again e (i.e itself)
First derivative of e = e
Second derivative of e = e
nth derivative of e = e
Therefore e^x is the function whose derivative of any order (first, second, third, …nth derivative) is equal to the function itself.
Now let see 2nd option e^-x. Its first derivate is -e^-x, 2nd derivative is e^-x, 3rd derivative is -e^-x, ...
Now let see 3rd option e^f(x). Its first derivate is e^f(x) . f'(x) and 2nd derivative is e^f(x) . f''(x), ...
Therefore e^x is the only function whose nth derivative is equal to the function itself.