Sum Rule is one of the Techniques of Differentiation.
The Sum Rule states that:
The Derivative of sum of two Functions is the Sum or Derivatives of the two functions.
Mathematically we can write this as:
If,
Or, function “h” is the sum of functions “f” and “g”
Then ,
Or. “function h” prime or derivative of function “h” is “func
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To simplify the process of Differentiating an equation or Function , There are some Rules or Techniques
using which we can simplify the process of finding the Derivative of any type of Function.
The main five techniques or rules of Differentiation are:
1> The Sum Rule.
2> The Product Rule.
3> The Power Rule.
4> The Quotient Rule.
5> The Chain Rule.
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The derivative and calculations on finding derivative of simple algebraic functions or polynomial functions is given below:
1> Derivative of Constant function or derivative of f(x)=y=c (c is a constant)
Let Δx be a small increment in x and Δy be corresponding increment in y. Then,
or,
and ,
Thus ,
2> Derivative of Identity function or derivative of f(x)=y=x
Let Δx
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Differential calculus or the concept of Derivative and Differential Coefficient was discovered by Isaac Newton (1642-1727) and Gottfried Wilhelm Leibnitz (1646-1716) in the process of solving two old problems one of finding slope of tangent drawn to a curve and another of finding instantaneous velocity of an object in non-uniform motion.
Derivative:
When a variable “y” is defined as a
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The basic theorems based on continuity are given below:
If the functions f(x) and g(x) are continuous at x=a then,
1> is continuous at x=a.
2> is continuous at x=a.
3> is continuous at x=a , if g(x) is not equal to 0.
4> is continuous at x=a if f(x) is greater than 0 , or is a positive number when “n” is even.
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A function “f” in interval [a,b] is said to be a continuous function when the Graph drawn for f(x) is a smooth line or curve without any break in it.
Such curve or line can be drawn by the continuous motion of a pencil in a sheet of paper.
And Discontinuous function is just opposite of the continuous function , the function “f” is said to be discontinuous function when
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The limit theorems or basic properties of limit are given below:
1> The limit of the sum (or difference) of the functions “f” and “g” is the sum (or difference) of the limits of the functions
i.e.
2> The limit of the product of the function “f” and “g” is the product of the limits of the functions.
i.e.
3> The limit of the quotient of t
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Let an Interval be denoted by (a-β , a+β) which is shown by the figure below:
and x∈ (a-β , a+β)
And let a function f(x) be defined at the Interval (a-β , a+β) .
Then we can also find the limit of function f(x) as,
Left hand Limit of a Function:
In the above case the limit of f(x) when “x” approaches “a” from the left hand side
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What is Interval?
A set of points lying between any two points “a” and “b” is known as an Interval.
For example:
the interval from points x=1 to x=10 is the set of points lying between 1 and 10.
Open and Closed Interval:
An Interval which includes it’s end points(a and b) is known as Closed interval ; While an Interval Which does not includes it’s end points(
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What is limit?
First study the following picture:
You can see in the figure ; if a polygon is inscribed inside a circle , then
as we increase the number of sides of polygon the area of polygon gradually increases .
As we increase the number of sides in the polygon it’s area gradually approaches
the area of circle and if we take a polygon with sufficiently large number of sides in same con
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