where v is the instantaneous velocity of the stone at time x. For example, the
instantaneous velocity of the stone after 1 second (when x = 1) is 16 × (2 × 1)
= 32 feet per second; after 3 seconds it is 16 × (2 × 3) which gives 96 feet per
second.
If we compare Galileo’s distance formula y = 16 × x^2 with the velocity formula
v = 16 × (2x) the essential difference is the change x^2 to 2x. This is the effect of
differentiation, passing from u= x^2 to the derivative = 2x. Newton called = 2x
a ‘fluxion’ and the variable x a fluent because he thought in terms of flowing
quantities. Nowadays we frequently write u = x^2 and its derivative as du/dx = 2x.
Originally introduced by Leibniz, this notation’s continued use represents the
success of the ‘d’ism of Leibniz over the dotage of Newton’.
The falling stone was one example, but if we had other expressions that u
stood for we could still calculate the derivative, which can be useful in other
contexts. There is a pattern in this: the derivative is formed by multiplying by the
previous power and subtracting 1 from it to make the new power.