Definition:
The derivative of a function f at a number a, denoted by f′(a), is the value
f′(a)=limx→af(x)−f(a)x−a,
if this limit exists.
Equivalently,
f′(a)=limh→0f(a+h)−f(a)h.
Besides being a rate of change, f′(a) is the
slope of the tangent line to y=f(x) at (a,f(a)), and is the
conversion factor from small changes in x to small changes in f(x).