Differentiation Rules and their Significance

Differentiation is a mathematical technique for determining the derivative of a function involving two independent variables. This procedure is designed to determine the instantaneous changes that occur in one of the variables as a result of the changes that occur in the other variable. For example, the instantaneous change in the rate of displacement when time is taken into consideration is referred to as velocity.

If we elaborate the process, the changes in variable ‘y’ with regard to another variable ‘x’ are expressed as dy/dx, where dy is the difference between the two variables. If y = f(x), then f'(x) is equal to dy/dx. This is the derivative of the function f(x).

Derivative Rules or differentiation rule

It is necessary to do a proper examination of the derivatives. In fact, the result of differentiating a function will be universal in application. As a result, there are some laws or principles of differentiation that you must comprehend and observe. Take a look at the list of such rules that are provided below.

•Power Rule of derivatives

This is one of the fundamental rules of differentiation that you will find less difficult to comprehend than the others. Consider the changes that occur in a function as a result of the application of power rules.

If f(x) = xn, 

then f'(x) = d/dx (xn) = nxn-1. 

If we take an example, you will have a better understanding of the application.

If f(x) = x6, then

Then, d/dx (x6) = 6x6-1 = 6x5, which is the solution.

•Sum Rule of derivatives

A function represented by the difference or sum of two smaller functions is subject to the sum rule of derivatives, which recommends that the following modifications should be made.

If f(x) = m(x) ± n(x)

Then, f’(x) = m’(x) ± n’(x)

This formula demonstrates that the signs of the smaller functions will be preserved, but that these functions will be subjected to the rules of derivatives.

Consider the following illustration.

If f(x) = x2+ x3 then

Then f'(x) = 2x + 3x2 is obtained.

This is an example of how the derivatives sum rule is implemented.

•Product Rule of derivatives

As a result of this rule, if the function of a variable is the sum of the products of two other functions, the following is the result.

If f(x) = m(x) × n(x),

Then, f’(x) = m′(x) × n(x) + m(x) × n′(x)

Take a look at this illustration to better grasp this concept.

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

= 5x4

This will be the outcome of the derivatives regulation in question.

•Derivatives of the Quotient Rule

The quotient rule derivatives provide guidance on how to execute a differentiation of a function when there are two terms in the division mode of the function. Here is what the rule offers as a starting point.

If f(x) = m(x) / n(x),

Then, f′(x) = m'(x)×n(x)−m(x)×n'(x)/ (n(x))2

After conducting differentiation, if you adhere to the rule and enter the values of the functions, you will receive an exact answer.

•Derivation of Chain Rule 

Suppose that a function is represented by a function with another variable, and that the variable of the first function is represented by a variable of the second function. The derivation of the chain rule suggests that the following differential operation be performed.

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 relatively simple to put into action if you pay attention to the phrases in each stage and learn the ways that are used to prove the chain rule.

Conclusion

When two independent variables are involved in a function, differentiation is a mathematical technique for calculating the derivative of the function. In this process, the changes that occur in one of the variables as a result of the changes that occur in the other variable are calculated at a given point in time.If we elaborate the process, the changes in variable ‘y’ with regard to another variable ‘x’ are expressed as dy/dx, where dy is the difference between the two variables. If y = f(x), then f'(x) is equal to dy/dx. This is the derivative of the function f(x).It is necessary to do a proper examination of the derivatives. In fact, the result of differentiating a function will be universal in application. As a result, there are some laws or principles of differentiation that you must comprehend and observe.