It should be clear that if two functions differ by a constant value, then they have the same derivative. In other words, if [math]f[/math] and [math]g[/math] are two functions and there exists a number [math]c[/math] such that [math]g(x)=f(x)+c[/math], then [math]f'(x)=g'(x)[/math]. The surprising result is that the converse is true, too:[br][br]If [math]f'(x)=g'(x)[/math] at each point [math]x[/math] in an open interval [math](a,b)[/math], then there exists a constant [math]c[/math] such that [br][math]g(x)=f(x)+c[/math] for all [math]x[/math] in the open interval [math](a,b)[/math]. [br][br]Thus, the [i]only way[/i] 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 [math]x[/math]-value, the slopes of the tangent lines are the same.[br][br]In this interactive figure, you can adjust the [color=#980000][b]value of [/b][math]c[/math][/color] and move the[b] gray point[/b] along function [math]g[/math].

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]