Example 2

Given y + √y = x², find dy/dx.

This answer can be used to find the slope at any point on the curve, For example, at the point (√2, 1) the slope is

while at the point (-√2, 1) the slope is

To get dy/dx in terms of x, we solve the original equation for y using the quadratic formula:

Since √y ≥ 0, only one solution may occur,

Then

The graph of this function is shown in Figure 2.8.1. By substitution we get

Often one side of an implicit function equation is constant and has derivative zero.