Example 2

Find dy/dx where x = y⁵ + y³ + y. Compute dy/dx at the point (1, 3). Although we cannot solve the equation explicitly for y as a function of x, we can see from the graph in Figure 2.4.6 that there is an inverse function y = f(x).

By the Inverse Function Rule,

This time we must leave the answer in terms of y. At the point (1, 3), we substitute 3 for y and get dy/dx = 1/433.

For y ≥ 0, the function x = yⁿ has the inverse function y = x^1/n. In the next theorem, we use the Inverse Function Rule to find a new derivative, that of y = x^1/n.