alt="images"/> plays the role of a parameter. If is slightly changed to (with small), the new root will accordingly move to a new value . If the conditions of the inverse function theorem are satisfied in a neighborhood of , the equation defines as a function with , where is the inverse function of . Accordingly, the derivatives of at and of at satisfy the relation
(1.21)
or for short, where the prime symbols acting on and must be understood as derivatives with respect to and , respectively. We can now estimate the location of the new root. Since is small,
Since the new root is precisely located at ,
Finally, since , we conclude from the last equation that . Taking the absolute value and recalling that ,
This last expression predicts how far the new root will move in terms of how much we have perturbed . From (1.22), we see that no matter how tiny may be, can potentially be very large if is very small. As an example, take the equation . For , the previous equation has a simple root at and a double root at (see Figure 1.3b, black curve). For and , the simple root exhibits a slight displacement to and , respectively (see dashed and solid gray curves in Figure 1.3b, respectively). However, for the same values of , the double root does experience remarkable changes. In particular, for , the double root disappears, leading to two simple roots located at and . For the effects are even more drastic since the function has no longer any root near .
alt="images"/> plays the role of a parameter. If 
