’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: