Partial Derivation of the Differentiation Formula - for reference
only - do not learn
Introduce δ to mean a very small change (Δ means a large change)
Gradient of the chord AB =
Now δy = y(x + δx) - y(x)
gradient of the tangent at A = 
Example y = x2.
If y(x) = x2
y(x+δx) = (x + δx)2 = x2 +2x.δx + (δx)2
gradient of tangent = 
= 
= 2x + δx
which tends to 2x as δx tends to 0
Higher powers of x: If y = xn we need (x + δx)n - xn
Pascal's triangle gives the multipliers of (x +a)n, and it can be shown that this leads to the 'magic' formula:
(xn) = nxn-1