Linearity of differentiation
Encyclopedia
In mathematics
, the linearity of differentiation is a most fundamental property of the derivative
, in differential calculus
. It follows from the sum rule in differentiation
and the constant factor rule in differentiation
. Thus it can be said that the act of differentiation is linear, or the differential operator
is a linear operator.
Let f and g be functions, with
and 
fixed. Now consider:
By the sum rule in differentiation
, this is:
By the constant factor rule in differentiation
, this reduces to:
This in turn leads to:
Omitting the bracket
s, this is often written as:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the linearity of differentiation is a most fundamental property of the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
, in differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
. It follows from the sum rule in differentiation
Sum rule in differentiation
In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist. This is a part of the linearity of differentiation. The sum rule in integration follows from it...
and the constant factor rule in differentiation
Constant factor rule in differentiation
In calculus, the constant factor rule in differentiation, also known as The Kutz Rule, allows you to take constants outside a derivative and concentrate on differentiating the function of x itself...
. Thus it can be said that the act of differentiation is linear, or the differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
is a linear operator.
Let f and g be functions, with
and fixed. Now consider:By the sum rule in differentiation
Sum rule in differentiation
In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist. This is a part of the linearity of differentiation. The sum rule in integration follows from it...
, this is:
By the constant factor rule in differentiation
Constant factor rule in differentiation
In calculus, the constant factor rule in differentiation, also known as The Kutz Rule, allows you to take constants outside a derivative and concentrate on differentiating the function of x itself...
, this reduces to:
This in turn leads to:
Omitting the bracket
Bracket
Brackets are tall punctuation marks used in matched pairs within text, to set apart or interject other text. In the United States, "bracket" usually refers specifically to the "square" or "box" type.-List of types:...
s, this is often written as: