A function has a derivative at a point [math]a[/math] if the slopes of the secant lines through [math]P\left(a,f\left(a\right)\right)[/math] and a nearby point [math]Q[/math] on the graph approach a finite limit as [math]Q[/math] approaches [math]P[/math]. If the secants fail to take up a limiting position or become vertical as [math]Q[/math] approaches [math]P[/math], then the derivative of [math]f[/math] does not exist at [math]a[/math]. [br][br]The interactive figure here demonstrates several cases where a function does not have a derivative at a particular point.