Studying calculus is an important part of the mathematical skill development of the students. These concepts of differential and integral calculus will be used in various domains of higher studies. Hence, learning the basic and advanced concepts of differentiation rules will create a strong foundation among the students. It will help to grasp the topics of the subjects related to higher mathematics and science applications in the professional courses. Here is the proper elaboration and explanation of the derivative rules you need to understand and learn to apply and solve problems.
What is Differentiation?
In the previous classes, you have studied the different functions that contain two variables. In these functions, a variable depends on the values of the other variable. The relation between these variables is interpreted using a formula/function/mathematical expression. This expression can be algebraic, trigonometric or related to any domain of mathematics.
Differentiation is the mathematical way to find the derivative of a function of two variables. This process is developed to find the instantaneous changes occurring in one of the variables depending on the changes of the other one. For instance, the instantaneous change in the rate of displacement considering time as the prime factor is called velocity.
If we elaborate the process then the changes in variable ‘y’ with respect to another variable ‘x’ is expressed as dy/dx. If y = f(x) then, f’(x) = dy/dx. Here, f’(x) represents the derivative of f(x). There are differentiation rules you will study in the Class 12 Maths syllabus so that you can easily carry out these operations on the functions given.
Differentiation or Derivative Rules
The evaluation of the derivatives should be properly. In fact, the result coming out of differentiating a function will be universal. Hence, there are some differentiation laws or rules that you need to understand and follow. Check the list of such rules mentioned below.
1. Power Rule of Derivatives
This is one of the basic rules of differentiation that you will find easier to understand. Observe the changes in a function when a power rule is applied.
If f(x) = xn,
Then, f’(x) = d/dx (xn) = nxn-1
If we consider an example, you will understand the application properly.
If f(x) = x6
Then, d/dx (x6) = 6x6-1 = 6x5
2. Sum Rule of Derivatives
If a function is represented by the difference or sum of two smaller functions, the sum rule of derivatives suggests the following changes.
If f(x) = m(x) ± n(x)
Then, f’(x) = m’(x) ± n’(x)
This formula shows that the signs of the smaller functions will be retained but these functions will follow the derivative rules. Consider this example.
If f(x) = x2 + x3
Then, f’(x) = 2x + 3x2
This is how the sum rule of derivatives is executed
3. Product Rule of Derivatives
According to this rule, if the function of a variable is the product of two other functions, then the outcome will be as follows.
If f(x) = m(x) × n(x),
Then, f’(x) = m′(x) × n(x) + m(x) × n′(x)
Consider this example to understand this rule better.
If f(x) = x2 × x3
Then, f’(x) = d/dx (x2 × x3)
= x3 × d/dx (x2) + x2 × d/dx (x3)
= x3 × 2x + x2 × 3x2
= 2x4 + 3x4
This will be the outcome of this rule of derivatives.
4. Quotient Rule Derivatives
The quotient rule derivatives suggest how to perform a differentiation of a function where there are two terms in division mode. Here is what the rule suggests.
If f(x) = m(x) / n(x),
Then, f′(x) = [m′(x) × n(x) − m(x) × n′(x)] / (n(x))2
If you follow the rule and put the values of the functions after performing differentiation, you will get the accurate answer.
5. Derivation of Chain Rule
If a function is represented by a function with another variable and this function is represented by the variable of the first function, then the derivation of chain rule suggests the following differential operation.
If f(x) = m(u) and u = n(x),
Then, f’(x) = d/dx f(x) = d/du m(u) × d/dx n(x)
This formula or rule is quite simple to execute if you observe the terms properly in every step and learn the approaches proving the chain rule.
Learning Differentiation Rules is Fun
Take one step at a time and cover every rule related to the derived fractions. If you look carefully, you will find that the rules are nothing but the representation of the simpler rules such as sum, power, and product of derivative functions. It means that the basic rules are what you need to understand and then proceed to the next ones.
If you follow the rules of differentiation as elaborated in this section, it will become a lot easier to comprehend these concepts. The first segment of calculus will become much easier to understand and study. Your confidence will increase when you understand the basic differentiation rules and use them to execute the operations as required in the exercise sums. Keep practicing after learning these differentiation rules and solve problems.