 # Differentiation by First Principles

 From: Martin Bland Date: 28 August 1999 Subject: Differentiation I have a question to differentiate y=x2 by first principles. The answer is 2x, isn't it? But what does the by first principles bit mean?

### Maths Help suggests:

The question says you should differentiate  y = x2 by first principles.

This means you are not allowed just to write down the answer using any
'short-cut' rules, but must use the 'official' method as follows: First, consider a sketch of the graph of  y = x² (right).

You have to find an expression for , which represents
the gradient of the graph at the point P.

Point P has coordinates (x,x²) and
Q is a nearby point with coordinates (x + h,(x + h)²).

The line through P and Q has almost the correct gradient. So Now this answer of  2x + h is almost the correct answer for .

As Q 'slides down' to get closer and closer to P:

• The gradient gets closer and closer to the value of ;
• The value of h get closer and closer to zero.

We write: This is the answer you expected, but done without reference to any 'short cut' rules!