impDif(equation, independent variable, dependent variable)
For instance, to take the derivative of xy = 4 you would use:
impDif(x*y=4,x,y)
As far as I know, the TI89 will not perform implicit differentiations directly, but it is relatively easy to program a userdefined function to do the job. It does take, however, a little knowledge beyond elementary calculus.
The idea is based on a theorem from multivariable calculus:
where z is a function of both x and y, and is the partial derivative of z with respect to x. This means "take the derivative as if x was the variable, and y (and any other symbol) was a constant." For instance, suppose we want to take the derivative (of y with respect to x) of:
xy^{2} + x^{2}y = 1
First, bring everything to the left side:
xy^{2} + x^{2}y  1 = 0
and then write the expression on the left in the form z = f(x, y):
z = xy^{2} + x^{2}y  1
Now, treating x as the variable in the function:
and treating y as the variable in the function:
so, by the theorem above:
You can verify that this result is correct.
impDer(equation, independent variable, dependent variable)
(Refer to your user's manual pages 265266 and 831.)
Fortunately, the method described above can easily be automated, because when you ask the TI89 to take a derivative of a function with respect to a variable, it actually takes a partial derivative of multivariable expressions.
The implicit differentiation function for the TI89 is:
Define id(f) = d(f, x)/d(f, y)
Note: This function assumes that the independent variable is "x" and the dependent variable is "y."
Here are the derivatives that were shown above, taken on the TI89. Notice that you have to be careful to type "x*y" instead of just "xy"  the latter expression is taken to mean the variable named xy and not the product of x and y. 

I called my function "id" for "implicit derivative". It is defined as shown. 

Here is the solution to the problem discussed above. 

The function shown above is convenient, but it will only find dy/dx. If you want to enter functions of other variables or find other derivatives, you will need to expand your function to enter the independent and dependent variable as shown. 

For example, "If sin(rt) = r, what is dr/dt?" Of course, you could convert to x's and y's, (and then convert back to r's and t's!) or solve the problem as shown at right. You should verify (by hand) that this is the correct result. 
