Differentiation of the sum

Instead of using standard ways of differentiation, the differentiation rules make it easy to assess derivatives of certain functions. The property of linearity has a notable relation to the method of acquiring the derivative of a function. The derivative sum of two or more functions is to be calculated in some situations in differential calculus. It is impossible to detect this derivative straight away. But it is possible to find it by the sum of its derivatives using the equivalent mathematical operation. Let us now master in detail the differentiation rules with appropriate examples for each of them below.

Differentiation rules:

Before beginning with how the rules of differentiation work, let us learn the different rules of differentiation:

1. Chain rule
2. Sum and difference rule
3. Power rule
4. Quotient rule
5. Product rule

We will now discuss these rules in detail.

Power Rule of Differentiation

In calculus, the power rule of differentiation is defined as the rule of derivatives that assist in finding a variable’s derivative that is raised to a power. Examples of these variables are x^2, 4x^5, or 2x^8. In this rule, the way to find the derivative of these variables is to multiply the exponent by the coefficient and then subtract the exponent by 1.

Example of the power rule of differentiation:

Find the derivative of y = 6x^5

In this question, the power of the variable is= 5

Now, multiplying the coefficient with the power, we get: 5 x 6 = 30

Reducing the power by one number, we will get = 4

Therefore, our solution is,

dy/dx = 30x^4

Sum Rule of Differentiation

Let us now understand how the sum rule of differentiation works. In this situation, if a function is either a sum or difference of two functions, then we come to the conclusion that the function’s derivative is either the sum or difference of each function. Simply put, this rule says that the derivative of a sum and the sum of a derivative are equal. 

Example of the sum rule of differentiation:

Find the derivative of f(x) = x^2 + x^3

Using the sum rule of differentiation, the equation becomes,

f’(x) = d/dx(x^2 +x^3)

f’(x) = d/dx(x^2) + d/dx(x^3)

f’(x) = 2x + 3x^2

Product Rule of Differentiation

The product rule of differentiation helps us find the product of a derivative. In this rule, if we say that y is a product of two different functions of u and v, 

The product rule thus becomes,

y = uv

After using the product rule, we get the equation,

dy/dx = u(dv/dx) + v(du/dx)

Example of the product rule of differentiation:

Find the derivative of x.cos(x)

Let us equate cos(x) as f(x), and x as g(x)

Now differentiate both of these individually,

f’(x) = -sin x

And g’(x) = 1

We get,

{ f(x)g(x) }’ equals to { g(x)f’(x) + f(x)g’(x) }

{ f(x)g(x) }’ = { -x sin x + cos x }, which is our solution

Quotient Rule of Differentiation

The quotient rule of differentiation is defined as the rule in which a quotient’s derivative is considered to be equal to the denominator times the numerator’s derivative that has been subtracted by the numerator times the denominator’s derivative. In the end, divide the whole of it by the denominator’s square. This rule is given as follows,

Consider a f(x) = u(x)/v(x)

Therefore, this function becomes,

f’(x) =( u’(x)  v(x) – u(x)  v’(x))/[v(x)]^2

Example of the quotient rule of differentiation:

Find the derivative of x^2/(x+1)

To find the solution of this function, let us equate x^2 with u(x), and (x+1) with v(x)

Therefore, the equation becomes, 

f’(x) = [v(x)u’(x) – u(x)v’(x)]/[v(x)]^2

So,

f’(x) = (2x^2 + 2x – x^2)/(x + 1))^2

Therefore, our solution is,

f’(x) = (x^2 + 2x)/(x + 1)^2

Chain Rule of Differentiation

The chain rule of differentiation explains how to find the derivative of a composite function. For the majority of the part, this rule is used in the substitution method to carry out the differentiation of composite functions. This rule is given as follows,

Consider a f(x) = u(v(x))

Therefore, this function becomes,

f’(x) =( u’(v(x)) v’(x))

Example of the chain rule of differentiation:

Find the derivative of sin 2x.

Using the chain rule, we will find the function’s derivative.

Now let us equate sin 2x with f(x), and 2x with g(x).

Through the chain rule, we come to a conclusion that the derivative of

sin 2x = cos 2x . 2

Which is further = 2 cos 2x (our solution).

Conclusion:

In this article, we learnt in detail the different rules of differentiation. These rules will help you find the derivative of functions under different situations. We got to learn about the rules of differentiation, including the power rule of differentiation, sum or difference rule of differentiation, quotient rule of differentiation, and chain rule of differentiation. Along with the definitions and workings of these rules, various examples of each of the rules is mentioned above in the article. The formulas of all the rules have also been informed in this article.^