An Implicit Differentiation Problem from MaplePrimes

On November 22, Joe Riel posted an implicit differentiation problem that caught my attention. It took the manipulations typically learned in an Advanced Calculus course one step further, but the devices learned in such a course could readily be applied. Joe's solution was expressed in terms of exterior derivatives and exterior products, so he used the liesymm and DifferentialGeometry packages to obtain solutions.

Here's the problem: Given a constrait, , and functions  and , find .

Here's a solution that a student in an Advanced Calculus course could be expected to fathom.

Define  and  and assume that the inverse function theorem allows us to write  and . The constraint equation then becomes . From the derivative sought, infer that  can be obtained from . Hence, from  we obtain 

or

where the derivative  is the required . To obtain  and , return to the constraint

and apply the chain rule, obtaining

and

To obtain , write out at length the inverse functions 

and

and apply the chain rule, differentiating the first equation with respect to  and . Solve the resulting set of simultaneous equations with Cramer's rule, and recognize the determinant in the denominators as the Jacobian.

Repeat these calculations with the second equation:

Put all this together to get

which is what Joe Riel obtained. Of course, prior to working out the details that Joe merely alluded to, I worked a simple example to clarify the dependencies between the variables. Hence, take  as linear functions,

so that  and  are

The constraint then becomes 

Set this equal to zero and solve explicitly for , obtaining

Differentiate to obtain , that is, :

Now, apply the formula .

 =

The results agree.

Download implicit_diff.mw