Chapter 4: Partial Differentiation
Section 4.3: Chain Rule
Example 4.3.24
If the equation implicitly defines , obtain .
Solution
Mathematical Solution
Obtain by differentiating the identity so that .
Now differentiate as if it were a function . The chain rule gives , so that
Note the use of the quotient rule for differentiation, the replacement of with , and the assumption of equality of mixed partial derivatives.
Maple Solution - Interactive
Obtain
Write the equation ; Press the Enter key.
Context Panel: Differentiate≻Implicitly Complete top portion of dialog as per Figure 4.3.24(a)
Context Panel: Expand≻Expand
Context Panel: Conversions≻to diff notation
Figure 4.3.24(a) Implicit Differentiation dialog
Careful scrutiny reveals that this last expression is equivalent to
a form that can only be approximated in Maple output upon the invocation of functions from the Typesetting package.
Maple Solution - Coded
Initialize
Simplified Maple notation is available if the commands to the right are first executed.
Apply the implicitdiff command, and temper the result with expand and a convert
The result without the conversion of notation
The result without the expand and convert operations
The best output without the notational advantages of Typesetting
Remove the Typesetting notational improvements.
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