8 Techniques of Differentiation
Differentiation is a relatively straightforward side of calculus. We only have to remember
a dozen or so standard derivatives (the elementary functions), a few rules (sum, product,
quotient, function of a function) and the rest is routine.However,wedoneedtobehighly
proficient with differentiation, able to apply the rules quickly and accurately. One reason
for this is to help you withintegration. One of the difficulties with integration lies in
recognising what was differentiated to give you what you have to integrate – you will
more easily spot this (and check it) if your differentiation is good.
While it is perhaps not essential for the engineers, I have includeddifferentiation from
first principles. It is really not that difficult to follow, and the ideas and skills involved
are useful elsewhere.
Prerequisites
It will be helpful if you know something about:- the idea of a limit (126
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)- basic algebra (Chapter 2
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)- properties of the elementary functions (Chapter 3
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)- the gradient of a line (210
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)- function notation (90
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)- composition of functions (97
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)- implicit functions (91
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)- parametric representation (91
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)Objectives
In this chapter you will find:- geometric interpretation of differentiation
- differentiation from first principles
- the standard derivatives
- rules of differentiation:
–sum- product
- quotient
- function of a function
- implicit differentiation