Fractional calculus is a generalization and extension of classical calculus. The development of calculus is usually attributed to Gottfried Leibnitz and Isaac Newton, although the two men had a bitter lifelong dispute over which of them developed calculus. Leibniz published his first work on calculus in 1684. Newton used calculus in his Principia, published in 1687. Although some of the main concepts of calculus were known to Greek, Arab, and Persian mathematicians for centuries, it was Leibnitz and Newton who created a complete framework and established the basic structure for calculus. The theorem of calculus connecting derivatives and integrals.

The derivative of a function, as a representation
d
f
d
x
or f′, is defined as the limit, if it exists, of the difference quant:
d
f
d

x

lim
Δ
x
→
0
Δ
f
Δ
x
, while the (indefinite) integral is defined as a limit of sum.
∫
a
x
f
u
d

u

lim
∑
Δ
f
Δ
x
, where the sum is extended to all Δx that make up the distribution of the interval (a, x).

A function F(x) is called an antiderivative (or primitive function) of f(x) if:
d
F
x
d

x

f
x
. A fundamental theorem of calculus states that the indefinite integral
F

x

∫
a
x
f
u
d
u
f(x) is a primitive function and that the definite integral can be calculated as
∫
a
b
f
u
d

u

F

b

F
a
, where F is a primitive function of f. Integration is the reverse process of differentiation.