Sum rule in differentiation
Encyclopedia
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. The rule itself is a direct consequence of differentiation from first principles
.
The sum rule tells us that for two function
s u and v:
This rule also applies to subtraction and to additions and subtractions of more than two functions
given by the sum
of two functions
u and v, such that:
Now let y, u and v be increased by small increases Δy, Δu and Δv respectively. Hence:
So:
Now divide throughout by Δx:
Let Δx tend to 0:
Now recall that y = u + v, giving the sum rule in differentiation:
The rule can be extended to subtraction
, as follows:
Now use the special case of the constant factor rule in differentiation
with k=−1 to obtain:
Therefore, the sum rule can be extended so it "accepts" addition
and subtraction
as follows:
The sum rule in differentiation can be used as part of the derivation
for both the sum rule in integration
and linearity of differentiation
.
so
In other words, the derivative of any sum
of functions is the sum of the derivatives of those functions.
This follows easily by induction; we have just proven this to be true for n = 2. Assume it is true for all n < k, then define
Then
and it follows from the proof above that
By the inductive hypothesis,
so
which ends the proof of the sum rule of differentiation.
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, the sum rule in differentiation is a method of finding 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...
of a function that is the sum of two other functions for which derivatives exist. This is a part of the linearity of differentiation
Linearity of differentiation
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...
. The sum rule in integration
Sum rule in integration
In calculus, the sum rule in integration states that the integral of a sum of two functions is equal to the sum of their integrals. It is of particular use for the integration of sums, and is one part of the linearity of integration....
follows from it. The rule itself is a direct consequence of differentiation from first principles
First principles
In philosophy, a first principle is a basic, foundational proposition or assumption that cannot be deduced from any other proposition or assumption. In mathematics, first principles are referred to as axioms or postulates...
.
The sum rule tells us that for two function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s u and v:
This rule also applies to subtraction and to additions and subtractions of more than two functions
Proof
Let y be a functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
given by the sum
SUM
SUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...
of two functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
u and v, such that:
Now let y, u and v be increased by small increases Δy, Δu and Δv respectively. Hence:
So:
Now divide throughout by Δx:
Let Δx tend to 0:
Now recall that y = u + v, giving the sum rule in differentiation:
The rule can be extended to subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
, as follows:
Now use the special case of 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...
with k=−1 to obtain:
Therefore, the sum rule can be extended so it "accepts" addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
and subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
as follows:
The sum rule in differentiation can be used as part of the derivation
Derive
Derive may refer to:*Derive , a commercial computer algebra system made by Texas Instruments*Dérive, the spontaneous exploration of urban landscapes guided by aesthetic instinct....
for both the sum rule in integration
Sum rule in integration
In calculus, the sum rule in integration states that the integral of a sum of two functions is equal to the sum of their integrals. It is of particular use for the integration of sums, and is one part of the linearity of integration....
and linearity of differentiation
Linearity of differentiation
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...
.
Generalization to sums
Consider a set of functions f1, f2,..., fn. Thenso
In other words, the derivative of any sum
SUM
SUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...
of functions is the sum of the derivatives of those functions.
This follows easily by induction; we have just proven this to be true for n = 2. Assume it is true for all n < k, then define
Then
and it follows from the proof above that
By the inductive hypothesis,
so
which ends the proof of the sum rule of differentiation.