In Leaving Cert Maths we are often asked to differentiate from first principles. This means that we must use the definition of the derivative — which was defined by Newton/ Leibniz — the principles underpinning this definition are these first principles. You can follow the argument at the start of Chapter 8 of these notes:
https://jpmccarthymaths.files.wordpress.com/2010/07/lecture-notes.pdf,
to see where this definition comes from, namely:
(*)
If you are asked to differentiate from first principles you must use this formula. There are a number of basic functions which you are asked to differentiate. Chapter 9 of
https://jpmccarthymaths.files.wordpress.com/2010/07/lecture-notes.pdf,
details the ones that are examinable.
Derivative of Powers
I will present two proofs of the differentiation of here. One is inductive and is the one they usually ask for; i.e.
Using induction prove that the derivative of is
.
They could also ask you:
Using induction, or otherwise, prove that the derivative of is
.
The “or otherwise” refers to the second, alternative, proof, that uses the Binomial Theorem (brutally cut from the Project Maths curriculum but in the tables.)
Proof by Induction
Let for
. Let
be the proposition that:
.
Consider . In this case
. Now
(usually just saying this is O.K. but to be more careful say that
is a line of slope 1 hence the slope of the tangent is 1 — or maybe throw
into *). Also
hence P(1) is true.
Assume is true; that is, if
then
.
P(k+1)? Let . Now
. Now using the product rule (where we use P(k)):
.
That is P(k) implies P(k+1) is true.
Hence by induction P(n) is true for all and we are done
Alternative Proof
Let . Now
; so expanding using the Binomial Theorem:
terms,
where means that the terms have a
or
… or
part. Now
,
.
Now taking the limit as goes to o, all the
terms contain a
(or a higher power of
so all go to 0.), and we have
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October 25, 2011 at 8:02 pm
Leaving Cert Project Maths: Proofs for 2012 « J.P. McCarthy: Math Page
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