Differentiation of the sum

Introduction

Differentiation is the process of finding out the rate of change of a parameter. The rate of change of a parameter is known as its derivative. There are different differentiation rules set to find the differentiation of sum and product or functions. Differentiation rules are nothing but a set of formulae that make the differentiation of a function simpler and less-time consuming. By the time you finish studying this guide, you will have grasped the basic differentiation rules, differentiation of the sum, the product, the power rule, and more.

What are differentiation rules?

Differentiation is one of the most important mathematical operations that help scientists to deduce various formulae. 

Important formulae of differentiation rules:

Take a function, f(x), then the derivative of f(x) is:

Take g(x), then the derivative of g is

Both g and f are functions of x. They are both differentiable with respect to x. Now, a few differentiation rules are as follows:

• The derivative of a constant function is 0

So, if f(x)=k, f'(x)=0

• The derivative of a function multiplied by a constant is equal to the derivative of the function multiplied by the constant

Or, 

If y=af(x) where a is a constant,

dy/dx=a df(x)/dx

y’=af'(x)

• Sum rule of derivative is 

(d/dx)  (f ± g) = f’ ± g’

• Product rule of differentiation

(d/dx) (fg) = fg’ + gf’

• Quotient rule of differentiation 

Sum Rule of Derivatives

In differential calculus, the derivative of a sum of 2 or more functions may be needed to perform some complex calculations. Mathematically, it is not feasible to find the derivative of a sum of functions; but it can be done by differentiating the individual functions and then adding them up.

What is the Power Rule on rewriting the expression?

The power of a variable, n, can also be rational or fractional, and so the variable could have exponents, and these exponents are real numbers. 

Then, to find the derivative,

1. Move the exponent down in front of the variable
2. Multiply it by the coefficient
3. Decrease the exponent by 1

For clear understanding, let us go through the following examples:

Product Rule of Derivatives

The product rule shows us how to differentiate expressions that are the product of 2 other, basic, expressions:

Product principle in calculus is a formula to find the differentiation or derivative of a function given as the result of two differentiable functions. That implies we can use the product rule to get the derivative of a function of the structure given as f(x)·g(x), with the condition that both f(x) and g(x) are differentiable. The product rule follows the idea of final points and subordinates in differentiation.

In calculus, the product rule is used to determine the derivatives of any of the functions given in the form of a product is obtained by the product of any 2 differentiable functions. The product rule says that “the derivatives of a product of differentiable functions are equal to the addition of the product of the 2nd function with the differentiation of the 1st function and product of the 1st function with differentiation of the 2nd function.  

The formula of Product Rule 

The derivative of a product of functions can be determined by using the product rule. The product rule formula is given as,

here,

• f(x) = Product differentiable functions u(x) & v(x)
• u(x), v(x) = Differentiable functions
• u'(x) = Derivative of function u(x)
• v'(x) = Derivative of function v(x)

Differentiating trigonometric functions

Trigonometric functions are also differentiable in their domain. The derivative of each trigonometric function is as follows:

Conclusion

Differentiation is an important subfield of calculus that is usually used in different branches of science, physics, engineering, etc. It is important to get the hang of its different rules, such as the power rule, sum rule, product rule, quotient rule, etc., to understand the concept in detail.