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) = ex, f'(x) = ex

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

Kevin Palmer

Hi Kevin,

The only functions that satisfy f' = f are functions f(x) = C ex where C is a constant.

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

g(x) = f(x)/ex

Then you have

g'(x) = [f'(x)ex - f(x)ex]/e2x

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

This means that g(x) = C for some constant C. But then f(x) = C ex 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.


Go to Math Central