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Articles : Implicit Differentiation

If 4y​4​​−5x​2​​y​2​​+x​4​​=0, find ​δy​​δy​​.

This is a typical implicit differentiation problem. The solution shows that the derivative depends on the values of both x and y, but how do students understand this?
In this example, we extend the power of visualization to problems of this sort.

1. Though it will not perform implicit differentiation directly, the CAS does handle it in steps, as shown to the right.
   Hint: enter the expression first, to keep a copy handy.
   Use Copy to insert it in your subsequent work.
   Now we turn to the Advanced Graphing App to explore further.
   
2. Press I and tap on the Advanced Graphing icon
3. Enter the equation in S1
4. Press P to see the graph
   The graph appears to be the lines, y=x, y=-x, y=x/2, and y=-x/2. The CAS factor command gives us a way to verify this.
5. Press K to return to the CAS view.
6. Press b, tap 1 Algebra, and select 4 Factor. Then tap on the history and drag back up to your original expression. Tap on it to select it and then tap Copy.
   Press E to see the factors of our expression.
   
   The factors agree with our understanding of the graph. But if the graph consists of those four lines, then the derivative is limited to the values -1, -1/2, 1/2, and 1. How do we reconcile this with the rational expression we have for our derivative?
7. Return to CAS view (K) and tap simplif. Press c, and select (the Where command). This command is used to make substitutions.
8. Tap and drag the history until you see our derivative expression. Select it and tap Copy.
9. Move to the second box in the template and enter y=x
   Hint: use lowercase variables in the CAS
10. Press E to see the result.
11. Simply copy the previous input and edit it for the other substitutions (y=x/2, etc.)

In this example, we extended the power of visualization to include a polynomial in x and y. The graph of the polynomial led to a better understanding of its derivative and to a conjecture regarding its properties, and the power of the CAS let us prove the conjecture. HP Prime, with its CAS and the Advanced Graphing app, represents a new standard in classroom tools to explore mathematics, make conjectures, and pursue proofs.

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