[AM_20151127DTCR] The “Onion” Method for Differentiation

Question

Introduction

     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”). 

Reminders

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.

Heuristics Used

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!