Chapter 4: Partial Differentiation
Section 4.3: Chain Rule
Example 4.3.29
The composition of with , produces the function . Express and in terms of and .
Solution
Mathematical Solution
Start with the identity and form two equations by differentiating first with respect to , then with respect to . The resulting equations are
and
Solve these equations for and . One approach to solving these equations "by hand" is to write the system in matrix form and apply Cramer's rule. Thus, write
so that
are the desired solutions.
Maple Solution - Interactive
Assign the identity the name
Context Panel: Assign to a Name≻
Differentiate and solve for and
Calculus palette: Partial-derivative operator
Context Panel: Solve≻Solve for Variables≻ and (See Figure 4.3.29(a) for data entry.)
Context Panel: Simplify≻Simplify
Figure 4.3.29(a) Entering and
Extract from this display the solution
Maple Solution - Coded
Notational simplifications
These commands allow and to be represented by and ,respectively; and for derivatives to be written with subscripts.
Differentiate the equation with respect to and
Apply the diff command.
Solve the two equations for and
Apply the solve command.
Display the two solutions on separate lines
Invoke the print command, using the tilde (~) to apply the command to each item in , the set of solutions.
Replace the names and with and , respectively; display the results
Apply the eval, simplify, and expand commands to obtain the results.
Invoke the print command, using the tilde (~) to apply the command to each item in .
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