Taking an Implicit Derivative


Introduction

This document explains how to take an implicit derivative on the calculator.
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Information

Implicit derivatives are done with the DERVX and DERIV commands, the same as with normal derivatives. However, the equations must be formatted in a specific way. Functions of the dependent variable must be identified to be derivated correctly. Also, the equation must be set to zero.
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Example of taking an implicit derivative

This document will demonstrate how to take the implicit derivative of X*COS(Y)+Y^2=X^3as an example.
Following are the steps used to work this example:
  1. (This step is not done on the calculator.) Set the equation to zero. This would be written as follows: X*COS(Y)+Y^2-X^3=0
  2. When entering the equation into the calculator, Y must be identified as a function of X. To identify Y as a function, it must be written as Y(X) instead of simply Y. Also, the zero must be dropped from the equation. The equation should be entered into the equation writer as follows: X*COS(Y(X))+Y(X)^2-X^3
    NOTE: If Step 2 is not done, the calculator will treat Y as a constant value, treating the derivative of Y as zero instead of Y'.
  3. Invoke the DERVX command:
    1. Press [LEFT-SHIFT], then CALC.
    2. Select F1 (DERIV) for Derivatives and Integrals.
    3. Select the DERVX command.
      NOTE: If it does not return the derivative immediately, it may be necessary to use the EVAL key.
The following will be displayed.
-(3X^2+d1Y(X)*SIN(Y(X))*X-(COS(Y(X))+2*Y(X)*d1Y(X)))
Although it is not shown, this expression is set equal to zero. d1Y(X) refers to Y' or the first derivative of Y. d2Y(X) is the second derivative, and so forth.
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Solving the expression for Y

When solving the expression for Y', the SOLVE command can be used with the d1Y(X) variable. Assuming that the calculator is in RPN mode, press ENTER after getting the above answer. This will put the expression on the stack. Then, enter d1Y(X) onto the stack, and invoke the SOLVE command.
NOTE: Entering the d as a lower case letter is critical for this step.
The following answer will be displayed:
d1Y(X) = ( COS(Y(X))-3*X^2 ) / ( X*SIN(Y(X)) - 2Y(X) )
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Further information

When taking a derivative of an expression with respect to a variable other than X, the DERIV command is used. Information on DERVX and DERIV can be found on page 14-12 of the Advanced Users Guide (or 14-14 of the downloaded version). Information on the SOLVE command can be found on page 14-57 (or 14-61 of the downloaded version). Click here to download the Advanced Users Guide (http://www.hp.com/calculators/techsupport/graphing/49g_userguide/).
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