Example 2 (Continued)

Let

x = 3f + 1,        y = √x.

(i)

(ii)

The equation

with t as the independent variable is trivial. We simply cancel the dx's. But when dy/dx is computed with x as the independent variable while dx/dt is computed with t as the independent variable, the two dx's have different meanings, and the equation is not trivial.

Similarly, the equation

is trivial with x as the independent variable but not when t is the independent variable in dy and dx, while x is independent in dy/dx.

The Chain Rule shows that when we change independent variables the equations

remain true.