# Understanding the Quotient Rule in Calculus
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Introduction to the Quotient Rule
This article marks the tenth installment in our calculus series, focusing on the foundational concepts behind the quotient rule in differentiation. In the previous discussion, I examined the product rule. Now, we will build upon that knowledge.
Let’s consider two functions defined as follows:
u(x) = 6x² + a
v(x) = 2x + b
y(x) = (6x² + a)/(2x + b) = u/v
dy/dx = ?? [where a and b are constants]
As with our earlier exploration, u(x) and v(x) are distinct functions of the variable ‘x’. We derive y(x) by dividing u(x) by v(x). The pressing question is: how can we compute the derivative dy/dx? Let's delve into this.
Intuitive Approach to Finding dy/dx
In the previous analysis, we used an intuitive method when y(x) was the product of u(x) and v(x). We first differentiated u(x) with respect to ‘x’ and then v(x) in a similar manner. Finally, we multiplied these derivatives to find dy/dx.
However, this method does not lend itself well to division, as it is unclear how to divide the numerator by the denominator. Thus, we must adopt a different approach based on first principles.
Applying First Principles to Derive the Quotient Rule
Following our previous methodology, let’s increase ‘x’ by a tiny increment, ‘dx’. Accordingly, ‘y’ increases by ‘dy’, ‘u’ by ‘du’, and ‘v’ by ‘dv’ (where ‘dy’, ‘du’, and ‘dv’ are infinitesimally small). The updated expression is as follows:
Now, we can simplify the right-hand side of the equation by employing algebraic division (specifically, polynomial long division):
Continuing this division method, we can resolve subsequent remainder expressions:
The last term in the remainder is of second-degree smallness, allowing us to disregard it and focus on the quotient as our solution. Let’s continue simplifying the expression derived from the quotient:
This results in the final expression which articulates the quotient rule of differentiation.
Interpreting the Quotient Rule
Here’s a concise step-by-step overview of how to implement the quotient rule:
- Multiply the denominator by the derivative of the numerator.
- Multiply the numerator by the derivative of the denominator.
- Subtract the result from step 2 from the result of step 1.
- Divide the final result by the square of the denominator.
Now, let’s apply this understanding to our initial example:
We have effectively utilized the first principles to grasp and execute the quotient rule of differentiation.
Next Steps in Our Series
Moving forward, I plan to tackle additional example problems to reinforce the concepts discussed thus far.
Quotient Rule for Derivatives - This video provides a clear explanation of the quotient rule, showcasing its application and significance in calculus.
Calculus: The Quotient Rule for Derivatives - In this video, the quotient rule is explored in detail, making it easier to understand how to differentiate functions involving division.
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Reference and credit: Silvanus Thompson.
You can read the original essay here.