Product Rule

Dhristi JEE 2022-24

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!


This method is used for finding the differentiation or derivative of the function provided in the form of two different functions or products. This means to say that the students can apply the rule of product or Leibniz rule for looking for the derivative function. The product rule is followed by the derivatives and limit concept in differentiation directly. In the below explanation provided by Vedantu, you will be able to comprehend the proofs, formulas and examples in a descriptive manner. 


Vedantu always tries to offer the best subject matter to its students. In the introduction of the product rule, we have explained the concept thoroughly so that it directly goes to the head of the students. 


In the tech-savvy world, online coaching has become the best choice as they have a chance to select a curriculum according to their schedule. Vedantu also provides the flexibility to select the teachers as per their rating and you can also interact with the teachers. It is not so that we keep on offering lectures to the students but random or planned online tests are also there. This helps them test their brains and they can also explore their lacking area. 


Once students are aware of their lacking areas, we help them overcome those challenges and in this process, our aim remains to keep up their confidence. By the time the examination is coming nearer, the tutors start guiding regarding the previous year sample question and ask students not to panic. This is not so that Vedantu only focuses on the sharp students, we provide equal guidance to all of them. However, we sort the students according to their individual capabilities and give some extra time to those who are lagging behind in some or another way.  This is done to help them pace with the other students. 


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!

Vedantu is going to show it with examples by applying it in varied situations. This will act according to the general differentiation rule, and the concept where constant derivative remains equal to zero. A constant derivative is multiplied with a function that is equal to a constant multiplication by function derivative. The sum of derivatives remains equal to the sum of derivatives. Now, it is time for you to experiment with these rules with the help of Vedantu tutors. Believe it you are going to enjoy this session as it will provide you with an insight into how successful this rule is while applying it practically. 


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 derivation 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 Detail

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.


Conclusion

Thus, how is your experience of studying with Vedantu for the topic of products rule? Yes, it must be wonderful as you have got the detailed information offered by the Vedantu teachers. After the definition of the product rule, you were having an opportunity of getting abreast of differentiation rules, three functions as well as formulas. The best part you must have felt by having online sessions with Vedantu is that you are given ample chances to do all these equations practically. Considering the fact that mathematics preparation is not possible in the absence of practical sessions, we strike a balance between both of the sessions to bring maximum benefits to the students via Vedantu. 

FAQs on Product Rule

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. To apply this rule, make sure you get yourself versed with both descriptions as well as formulas.

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 in the offered form of two different functions and products. This rule is followed as per the derivatives and limits of the concept in terms of differentiation straightly. Thus, this way, you will be able to describe the product role according to calculus.

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)


The formula of the product rule is really simple but students make sure you do not feel overconfident while applying it otherwise you may go wrong with a minor error.

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. For applying basic differentiation, you need to have clarity regarding constant multiple rules, power rule, difference rule, and the sum rule. With this clarity, you will go for it correctly and confidently.

5. Why should I choose Vedantu for the product rule concept?

Vedantu has become a wonderful educational portal to accomplish the ever-growing educational needs of competitive students. And mathematics is a subject that is considered a key to open career avenues in the future as well. So, Vedantu designs the entire classroom for mathematics for maintaining knowledge enhancement and score as well. As per the specific requirements of the students, Vedantu maintains both online and physical modes as it offers online sessions with the flexibility of downloading the material in offline mode as well. Thus, you are going to get the best results with Vedantu.

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