Friday, November 27, 2015

[AM_20151127DTCR] The “Onion” Method for Differentiation


     The differentiation of the secant function is not taught directly as part of the Additional Mathematics syllabus.  It can be derived from known facts.  I first show the standard application of the (extended) Chain Rule for novices, and then show a more effective way of applying the chain rule, which I called the “Onion Method”.  This looks like peeling onions or unpacking Matryoshka dolls (“Russian dolls”)


Solution 1  (for beginners)

Solution 2  (a more expedient way)

Final Remarks
     When we peel onions, we peel from the outer layer inwards.  Likewise, when we have a composite function, we differentiate from the outer layer first, and then work to the inner layers.  Every time we differentiate a layer, we write down the changed layer and then copy and paste everything within that layer.  With regular practice, this should become second nature.
     For another example of the “onion”, take a look at the derivative of the arcsecant function.

H04. Look for pattern(s)
H05. Work backwards
H09. Restate the problem in another way
H10. Simplify the problem
H11. Solve part of the problem
H13* Use Equation / write a Mathematical Sentence

Suitable Levels
GCE ‘O’ Level Additional Mathematics
GCE ‘A’ Level H1 Mathematics
GCE ‘A’ Level H2 Mathematics (revision)
International Baccalaureate SL & HL Mathematics
* AP Calculus AB & BC
* other syllabuses that calculus
* anyone who loves to learn!

1 comment:

  1. Actually, instead of memorising this as a procedure to be followed, it would be good if students can figure out the procedure on their own. That is another level of thinking and learning. Can you think of a common mathematical difficulty, and figure out yourself a procedure for solving it in your own words?