Calculus comes with a lot of rules you must learn in order to fully understand it. One of these rules is the quotient rule, which should be applied when you are trying to find the derivative of a function that has division within it. It’s a simple rule to understand and after a few examples you will definitely get the hang of it.
To understand the quotient rule, you must know all of the basics about derivatives. If you really want to learn all there is to know about derivatives, then you should try the Udemy course Calculus 1 – Derivatives. Having a strong understanding of calculus can help you in various career fields, and it can even help you better utilize and learn programming.
What is the Quotient Rule?
In calculus, it may be difficult to believe, but many of these rules are designed to help make the process of finding the derivative to these problems a lot easier. The quotient rule is no exception. Finding the derivative of a basic function is one thing, but when you start involving division and multiplication, additional rules have to be set to make the problems more manageable.
Simply put, the quotient rule is a formula that you use to find the derivative of a fraction. It actually uses two other rules, the product rule and the chain rule, which means if you learn those you will essentially know how to work out problems using the quotient rule as well.
Learning the Product and Chain Rule
When performing the quotient rule, you must know both the chain rule and the product rule, these two are apparent in the quotient rule, and it’s a great deal of help when you’re doing problems.
The chain rule is simple, and it is applied to composite equations that have layers to them. For example the equation (3x2 + 1)2 cannot be derived simply, and it would require the chain rule to be done properly. When using the chain rule, you take the derivative of the outer layer of the problem, which is in this case the 2 exponent.
This changes the problem to 2(3x2 + 1). After this is done you must then multiply this equation, but the derivative of the equation within the parentheses. So take the derivative of (3x + 1), which is equal to 3. And then multiply the two equations together.
Your result should be 2(3x2 + 1) (3) or 6(3x2 + 1). That’s all there is to it. The chain rule can help simply a lot of problems you may have if you use it whenever you can.
The product rule is rather easy to grasp as well once you get the hang of it. The definition of the product rule is that if two functions both have a derivative, then they can have a product. The formula for the product rule looks like (fg)’ = f’g + fg’.
What the formula is saying, is that in order to get the derivative of the product of both equations, you must take the derivative of one equation, multiply it by the second equation and then do the reverse and add the two products together.
An example can be demonstrated here with two very simple equations. Imagine f(x) = 2x and g(x) = 3x2. Now if you were to take the derivative of f(x) and multiply it by g(x), you would have 6x2 as your answer. If you were to do the same but take the derivative of g(x) and multiply it by f(x), you would have 12x2 as your answer. As you can see the two products of these equations are different, and when you add these two products together you get the derivative of f(x) and g(x) when they are multiplied together, which is also known as 18x2 in this particular instance.
Examples of the Quotient Rule
If the functions f(x) and g(x) have derivatives then the quotient has a derivative. The equation for finding the derivative of the quotient of these two functions is below.
When explained, the formula above simply states that you find the quotient of both functions by taking the derivative of the first function and multiplying it by the second function then subtracting it by the derivative of the second function multiplied by the first function, and finally dividing it all by the second function taken to the second power.
Let’s look at the previous equation we used before with f(x) being 2x and g(x) being 3x2. In order to find the quotient of these two equations we must first multiply them by their derivative. Luckily we already established the values of f’g and fg’ before when we did the product rule example. F’g equals 6x2 and fg’ equals 12x2. This already puts us a step in the right direction and it’s a good explanation of why the product rule is so important to learn.
Now we simply establish the fact that g(x) squared, also known as (3x2)2 is equal to 9x4. After applying the quotient rule our problem is 6x2 – 12x2 and it’s all divided by 9x4.
The rest of the problem is basic division and algebra. Your answer should be -2x6/3. See if you got it right and if not, take things back from the base equation and try it again.
The quotient rule is an integral part of calculus. Learn about this and other important parts of this type of math by checking out the Calculus 1 Essentials course on Udemy.
Understanding Higher Levels of Calculus
The quotient rule may seem like a hard concept to understand now, but it will become easy in due time. Calculus has higher levels of math for those that want to pursue it more and it involves other rules that makes use of the product, chain, and of course product rule. You will learn how these rules come in handy when you learn Calculus 2 or go so far as to study calculus 3. You should check out the Udemy courses that teach you all about Calculus 2 and Calculus 3 to the point where you can become a master at the subject.