Subject: Derivatives

Recently my calculus teacher asked his students to try and find any functions whose derivatives where the exact same as the original function.

The only function then I have determined that statement to be accurate in is all the natural exponential functions. Ex. f(x) = e^x, f'(x) = e^x

If possible could you please email me all the functions that you can find in which the original function and its derivative is identical.

Thanks,
Kevin Palmer

Hi Kevin,

For, let f be any function such that f' = f. define a new function g by the rule

Then you have

Since f' = f, g'(x) = 0

This means that g(x) = C for some constant C. But then f(x) = C e^x so f is a constant times the exponential function.

The fact that the functions with zero derivatives are the constant functions is a consequence of the mean value theorem: If g is a function that is not constant, and say g(a) = b and g(c) = d (with b different from d), then you can find some x between a and c such that

g'(x) = (d - b)/(c - a)

Therefore g cannot have a derivative of zero everywhere if the range of g contains at least two different values b and d.

Cheers,
Claude