Same derivative means functions differ by a constant
It should be clear that if two functions differ by a constant value, then they have the same derivative. In other words, if and are two functions and there exists a number such that , then . The surprising result is that the converse is true, too:
If at each point in an open interval , then there exists a constant such that
for all in the open interval .
Thus, the only way two functions can have the same derivative is if they differ by a constant value. Geometrically, this means that one function is simply vertically shifted away from the other; for any -value, the slopes of the tangent lines are the same.
In this interactive figure, you can adjust the value of and move the gray point along function .
Developed for use with Thomas' Calculus, published by Pearson.