Introduction
The product rule is one of the 2 main rules of differentiation you learn when studying calculus. This rule applies when you differentiate a function that has been formed by multiplying 2 (or more) functions of x together. This is a very useful rule of differentiation if you want to differentiate functions like xsin(x) or (3x+1)(x-5).
In next topic now we will try to elaborate the concept of product rule in simple easily understood words and will try to cover some of the worked out examples, highlight the most common errors and put together some of the most popular FAQ, so that for easy learning for students, teachers and self Learners.
What Is the Product Rule?
Product rule. The product rule is a rule of differential calculus for finding the derivative of a "product" of two functions. The product rule can be used to differentiate the function, where is the product of two functions and :
f(x) = u(x) v(x)
The derivative of f(x) is given by:
D/dx [F '] (x)=u ' ( x).v( x)+ u( x).v' ( x)
Which could be interpreted as: derivative of 1 st function meulsed by 2 nd function+1 st function meulsed by the derivative of 2 nd function.
Yeh that makes complete sense because I suppose using the plus/minus rule you could get the derivatives of the two then add them together. Using this rule the two are changing together.
Why Do We Need the Product Rule?
We are constantly working with functions that are in this form, in mathematics and in the natural world, revenue=price*quantity and in physics work=force*distance where two functions are multiplied, thus the derivatives of both of them were computed and whenever differentiating functions of this form we get the most famous formula called the product rule as follows:4.8].
We will have differentiated the polynomial before. But if the term is trig or exp or log then will be far too complicated or in fact impossible. So will differentiate the product rule:
Step-by-Step Method to Apply the Product Rule
Find the two functions u( x) and v( x) we are multiplying to:
Next, open the brackets and split the functions into 2 as follows: and now differentiate each one of them, u ( x) and v ( x):
This is the formula which you should be using below: f '(x)= u '(x)v(x)+ u(x) v' (x).
Defines the same term. Sy: sum of the term in s.
Some examples to illustrate this:
Worked Examples
Example 1: Simple Polynomial Product
Differentiate to findf ' ( x ) where
Let u (x) =x. Then u ' (x) =2x:
Define v(x)=x+3, then v0(x)=1
3. Now differentiating we obtain;f' x' = (2x) (x+ 3) +(x) (1) f' x' = 2x+ 6x+ x f' x' = 3x+ 6x
Example 2: Trigonometric Function
Differentiate f(x) = x sin(x)
Suppose u(x)=x, so that uprime(x)2x;
Let's define and then it is. And so That's definitely the last time I will write down the function so good luck remembering it!
F ( x). The next thing we will find is the value off' ( x ), with this we can findf ( 1 ).
Example 3: Exponential Function
What is the derivative of f(x) = e ln x?
If u is differentiable at 0; then we have that u' 0.
Let v ( x) =lnx, v'(x) = 1 / x.
And then after dividing by e on both sides we have, and after dividing by e on both sides, we have,
Example 4: Product of Three Functions
The product rule involves three functions, and if then.:
F' (x) = u' (x) v (x) w (x)+ u (x) v' (x) w (x)+ u (x) v (x) w' (x)
Then, add 3 functions, f, g and h. To add 3 functions, differentiate 2 functions and leave the third as it is. Do it for the other remaining 2 functions. Add the 3 derivatives.
Common Mistakes to Avoid
Differentiate from one line to another. 'Another common mistake is to' f (x) = u'(x) v'(x) ' No, you can't. We do have two factors here, like in the product rule:
Cannot even pulled alone we only indiated section of on student and no second term.
Sign mistakes,: Keep an eye on the minus signs whenever u(x) or v(x) appears.
No factorization: The unfactorized answer can be a bit of a handful in the later stages of a calculus problem (i.e. Finding the critical points).
Product Rule vs. Other Differentiation Rules
Here are some rules, which will show us how to get the product rule from these rules.
Sum/difference rule If 2 or more functions are adding/subtracting. DONOT multiply the functions. Find the derivatives of each terms.
Divide rule for a function g() by another function f() is precisely similar as product rule but in the denominator squared is:
The rule of composition is the differentiation of a composition of functions. I. E. A differentiating of a function of other function. The rule of composition is very often associated with the product rule when u (x) or v (x) are composite functions.
The most important skill, I think, involved in calculus is in knowing when a rule applies, and using more than one rule simultaneously when necessary.
Real-Life Applications of the Product Rule
This product rule is a piece of rules that are not horrible that you will learn from your teacher at school for instance it could be useful in the real world such as:
Physics AT by is used in dynamics where the quantity is not a constant but is actually a function of time (for example if the object has a varying mass and velocity). Change in a quantity, (for example the momentum, which is equal to the mass times the velocity).
E cons:change price (or Quantity) in respect of time or market, find the marginal revenue.
(Engineering: the branch of science concerned with the design and construction of systems wherein two or more variables are coupled. Such as, the pressure of a gas increasing as its volume decreases or two variables).
A very general model of population growth in which the time rate of change of the population size, d N/dt, can be non-zero.
Practice Problems
Try these on your own to test your understanding:
Subtractf(x) = ( 2x 1 )(x 4)
Differentiate f(x) = x cos(x)
Differentiate f(x) = x e
How does f(x)=(x+1)(x-1)(x+2) differ from one?
A few things for self evaluation 1. You may also like to test your understandings by going through all the steps below as self check with the step by step way as mentioned above and check your answer by expanding out all the polynomial forms (if it can expanded out,) and see whether it matches
Conclusion- Whenever we go to differentiate a product of two or more functions this might involve differentiation with product rule. To find differentiation of two products, make both the method; first block out differentiation of the two functions and then take recourse to the product rule. Second way: - Use the rule together with other differentiation rules; quotient rule, chain rule and addition of rules to get answers for the differentiations to be made.
FAQs
1- What products can be called in simplified form?
The products are called "first derivative of the first multiplied with second" "plus first times derivation of second
"2-How do you indicate the product?
Let u (x) v (x) then u '(x) v (x) + u (x) v' (x).
3-Can you make a rule for three products?
Surely if for three products we have f (x)= u(x)v(x) w(x) then as you probably know... Where f (x) the derivative of the product in two ways will again be an exact expression of it
.4-product rule and question rule;how they differ.
It will differentiate the multiplication of the two products, in contrast to the derivatives of division of the two products; it will be a different quotient with squared denominator
.5- what will happen when we find out the derivative first.
The product does not really do "distribute the differentiation" since "addition rule of differentiation does not apply for derivatives and therefore you are basically multiplying an extra solution with your differentiation and thus that result would be not appropriate". Because as was shown above the derivative has distributive-like behavior to a certain extent,but differentiation is not "adding up" which enables it to conduct "distributive operation".
6- The derivatives are also "independent " of one another so you will only get one solution to the function.
To do "derivatives and then product" and to try to add those derivatives is no better than to output the wrong addition from the simple polynomial multiplication, because you already have multiplied those derivatives to get two other solutions, which are "additional" but wrong! Hence: - Derivatives with help of the product rule will make sense to you when you work with very complex formulas. As can be seen for example. - The second case usage is applied generally when "polynomial expansion could be a pain or quite complicated"; that can be seen also
.7 - does it apply to common life?
In our daily life we use till we need to multiply two results of which the result depends on time: (for example: velocity x mass = momentum, quantity x prices = revenue).
Or some variable which depends on 2 things:- Chemistry, biology, physics, math, biology or economics, may occur in many cases. So for chemistry example if for a solution; the volume V is function of time
.8. -What are the usual mistakes students make differentiating with the help of the product?
Students, particularly most, in connection with two variables the product rule for working out the derivation; commonly differentiate their equations for each variable individually instead of taking both derivatives jointly. Or