Sum and Difference Rules of Differentiation

Some differentiation rules are a snap to remember and use. These include the constant rule, power rule, constant multiple rules, sum rule, and difference rule.

Differentiation meaning includes finding the derivative of a function. Every time we have to find the derivative of a function, there are various rules for the differentiation needed to find the desired function. 

What is Differentiation?

In general terms, differentiation in Mathematics is the process of finding the derivative of any function. In scientific terms, it is a rate of change in some of the functions. The basic step of finding the derivative of a function is taking a limit of difference. However, it becomes tedious to repeat every step. There are various rules for differentiation that will enable us to find the derivative to mitigate this. This article will discuss the fundamental theorems and formulas for the differentiation below. 

The First Principle of Derivative of Function:

The first principle of derivative of a function is “The derivative of a function at a value is the limit at that value of the first part or second derivative”. This principle defines the limit process for finding the derivative at a certain value because all functions have limits.

For example, consider

Consider  x = 4 and  y = x2. The latter has one more value than its parent: four. Hence, we need only one step from here to find that y = x . . . . (1) 

This principle seems simple and easy to understand, however, we need to be very careful throughout our study of differentiation.

Differentiation Rules

 The method of obtaining the derivative of a function or process of differentiation contains the property of linear functionality. This property makes the derivative more natural for functions constructed from the primary & elementary functions using the methods of multiplication & addition by a constant number. Let us discuss the differentiation rules with some of the examples in this article.

The formula above is for the case of a combination of two functions only. However, if more than two functions appear in the combination, the rules are practiced several times to give the derivatives. However, it can be noticed that the Sum and Difference Rules of Differentiation extend only once to give the derivative of the algebraic functions of number as the similar algebraic rules of the derivatives of the separate functions. 

The most important deduction from the (∗) and (∗∗) concerns the constants’ concepts in the derivation process. An additive(constant) can be regarded as the function of x that does not change the value as the x varies. This means that the derivative of the constant value is zero. Therefore, the additive constant does not count when the derivative is taken. 

To better understand why the constant is not considered to be a function during differentiation. Consider the following functions as the 

Where A is the constant and plots the figure that emerges in the figure, clearly revealing the derivative of the functions at every point that is the same, the only change is that the graph of the later function is A step away from the former function. 

Real-Life Applications of Sum and Difference Rule

There are distinct applications for the Sum and Differentiation in the field of Mathematics and other fields like engineering, science, physics, etc. In the whole article, we have learned different functions for derivatives like implicit functions, trigonometric functions, logarithm functions, etc. This is the most crucial topic covered in class 12 Mathematics anyone can learn from.

The concepts of derivatives can be large or small scales that are used in many ways, such as a change in velocity and change in temperature of different sizes and shapes of an object depending on the environment. However, the change in velocity is the most applicable for the derivative. For instance, to know the rate of change in the volume of dx and volume that represents the change in the sides of the cube. 

Other applications: 

• To calculate the loss and profit using the graphs in business. 
• To check the temperature variation. 
• To determine the distance or speed that is covered as kilometers per hour or miles per hour.
• To find the range of magnitude of the earthquake in the study of Seismology.

With the help of the Sum and Difference Rule of differentiation, we can find the rate of change of one quantity concerning another. Some of the instances are

• Acceleration: The rate of change of velocity concerning the time
• The derivative function is used to calculate the highest & lowest point of the curve in a graph or to know its turning point.
• To find tangent and normal to a curve.

Sum and Difference Rules of Differentiation:

With the help of the Sum and Difference Rule of Differentiation, we can derive Sum and Difference functions. These functions are used in various applications & each application is different from others. 

Sum Rule Definition: The derivative of Sum of two or more functions is equal to the sum of their derivatives.

Difference Rule Definition: The derivative of the difference of two or more functions is equal to the difference of their derivatives.

In simple terms, if the function has the sum or difference of two functions, then the derivative of the functions is the sum or difference of the individual functions.

If f(x)=u(x)±v(x), then;

f'(x)=u'(x)±v'(x)

Example 1: f(x) = x + x3 

Solution: 

Applying the sum rule of derivative formula here, we have:

f’(x) = u’(x) + v’(x)

Now, differentiating the given function, we get;

f’(x) = d/dx(x + x3)

f’(x) = d/dx(x) + d/dx(x3)

f’(x) = 1 + 3×2

Example 2: Find the derivative of the function f(x) = 6×2 – 4x.

Solution: 

Given function is: f(x) = 6×2– 4x

This is of the form f(x) = u(x) – v(x)

So by applying the difference rule of derivatives, we get,

f’(x) = d/dx (6×2) – d/dx(4x)

= 6(2x) – 4(1)

= 12x – 4

Therefore, f’(x) = 12x – 4

Example 3: If the function f (x) = x3 − 6×2 + ax + b satisfies Rolles theorem in the interval [1, 3] and f′ [(2√ 3 + 1) / √3] = 0, then find the value of a.

Solution:

f (x) = x3 − 6×2 + ax + b

f′(x) = 3×2 − 12x + a

f′(c) = 0

f′(2 + [1 / √3]) = 0

3 (2 + [1 / √3])2 −12 (2 + [1 / √3]) + a = 0

3 (4 + 1 / 3 + 4√3) − 12 (2 + [1 / √3]) + a = 0

12 + 1 + 4√3 − 24 − 4√3 + a = 0

a = 11

Conclusion

With the help of the Sum and Difference Rule of differentiation, we can find the rate of change of one quantity concerning another. The most important functions for derivatives are trigonometric functions, implicit functions, &  logarithmic functions, etc.