Differentiation By First Principles Essay, Research Paper

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.

Its gradient is 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 gets

closer and closer to zero. We write: