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