There are situations where a dependent variable is dependent on more than one independent variable. For example, the volume of a rectangular box depends on its length, width, and height, .

Think of a box shaped balloon being blown up.  The volume and all three dimensions are all changing at the same time. Their rates of change are related to each other.

Since rates of change are derivatives, all the derivatives are related. Given several of the rates, you can find the others.

In these problems you use implicit differentiation to find the relationship between the variables and their derivatives. That means that you differentiate with respect to time. And time is usually not one of the variables in the equations. – see no t anywhere. Really, the length, width and height are all functions of time; you just don’t see the t.

Sometimes the substitutions required to work your way through these problems are the tricky part. So, be careful with your algebra.

Course and Exam Description Unit 4 sections 4.3 to 4.5.