Product Rule

Product Rule - Differentiation Rules, Three Functions and Formulas

What is Differentiation?

Differentiation in maths, is the way of finding the derivative, or rate of change of some of the functions. The basic technique of the differentiation can be shown by doing algebraic manipulations. It has many of the fundamental theorems and formulae to do the differentiation of the functions. In this particular topic, we are going to discuss the basic theorems and some of the important differentiation formulas with suitable examples. Let us learn an interesting topic!


What is Derivative?

The derivative of a particular function can be defined as the rates of change of a function at that particular point.


What are the Differentiation Rules?

There are some basic product rule differentiation that you need to know!

1. The Sum Rule or Difference Rule

If function f(x) is a sum or difference of any two functions, then the derivative of the sum of any given functions is equal to the sum of their derivatives and the derivative of a difference of any given functions is equal to the difference of their derivatives.

Suppose, if we have a given function f(x), 

f(x)= u(x) ± v(x)

Then, the differentiation of function f(x), f'(x) =u'(x) ±v'(x)


2. Product Rule

According to the product rule differentiation, if the function f(x) is the product of any two functions, let's say u(x) and v(x) here, then the derivative of the function f(x) is,

If function f(x) =u(x) ×v(x) then, the derivative of f(x),

f′(x) =u′(x) × v(x) + u(x) × v′(x)


3. Quotient Rule

The quotient rule says that, if any function f(x) is in the quotient form or in the form of two functions [u(x)]/[v(x)], then the derivative of the function given function f(x)

If f(x) =u(x) v(x) then, 

The differentiation of function f(x), 

f′(x) = \[\frac{{u}'(x) \times v(x)-u(x) \times {v}'(x)}{v(x)^{2}}\]


4. Chain Rule

In chain rule, suppose a function y = f (x) = g (u) and if u = h(x), then according to product rule differentiation, dy dx = dy du × du dx .This rule plays a major role in the method of substitution which will help us to perform differentiation of various composite functions.


We Are Going to Discuss Product Rule in Details

Product Rule

Product rules help us to differentiate between two or more of the functions in a given function. If u and v are the two given functions of x then the Product Rule Formula is denoted by:

d(uv)/dx=udv/dx+vdu/dx

Whenever the first function is multiplied by the derivative of the second and the second function multiplied by the derivative for the first function, then the product rule is put up.  Here we take u as a constant in the first term and v as a constant for the second term.

The formula of the product rule seem like this for the product of the two functions.If we have a product of the three functions, then the formula can be written as following:


Three Functions

For three functions multiplied together, we get this:

(fgh)’ =  f’gh + fg’h  + fgh’

There is a pattern to this. Compare the two formulas carefully. Do you see how each of them maintains the whole function, but each term for the answer takes away the derivative of one of the functions? 


When is Product Rule Used?

Do you see how the f(x) is the product of the two smaller functions? We can also have a particular situation where the f(x) is the product for three or more of the smaller functions:

When you see functions such as this, then you can use the product rule.


A Few Differentiation Formulas and Examples Have Been Listed Below

Differentiation Formulas

If f(x) = tan(x)

f’(x) = sec2x

If f(x) = cos (x)

f’(x) = -sin x

If f(x) = sin (x)

f’(x) = cos x

If f(x) = ln(x) 

f’(x) = 1/x 

If f(x) = ex

f’(x) = ex

If f(x) = xn, where n is any fraction or any integer.

f’(x) = nxn-1

If f(x) = k , here k is a constant

f’(x) = 0


Product Rule for the Logarithms to Write an Equivalent Sum of the Logarithms

1) Factor the argument completely, by expressing each of the whole number factors as a product of their primes.

2) To write the equivalent expression by adding the logarithms for each of the factors.

FAQs (Frequently Asked Questions)

1. What is product rule maths?

The product rule is taken into account only if the two "parts" of the function are being multiplied with each other, and the chain rule is if they are being composed. For example, to find out the derivative of f(x) = x² sin(x), we use the product rule, and to find out the derivative of g(x) = sin(x²) we use the chain rule.

2. What is product rule in calculus?

The product rule is normally used for doing the calculus whenever we are asked to take the derivative of a given function that is the multiplication of a couple of or several smaller functions. In other words, a function

3. What is the formula of product rule?

Product rules help us to differentiate between two or more functions in a given function. If u and v are the given function of x then the Product Rule Formula is given by: If function f(x) =u(x) ×v(x) then, the derivative of f(x),

f′(x) =u′(x) × v(x) + u(x) × v′(x)

4. What is basic differentiation?

Some differentiation rules to be remembered and used. It includes the constant rule, the power rule, constant multiple rules, sum rule, and the difference rule. The constant rule: This is simple. f (x) = 5 is the horizontal line with a slope value of zero, and therefore its derivative is also known to be zero.