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

## 1 comment

Comments feed for this article

October 25, 2011 at 8:02 pm

Leaving Cert Project Maths: Proofs for 2012 « J.P. McCarthy: Math Page[…] Differentiate from first principles: , , , , , . […]