Using the Quotient Rule to Find the Derivative of a Complex Function

Welcome to this comprehensive guide on how to find the derivative of a complex function using the quotient rule. Specifically, we will tackle the function f(x) 3x-5 / 5x2. This process involves several steps and requires a keen understanding of the quotient rule.

Introduction to the Quotient Rule

The quotient rule is a powerful tool in calculus for finding the derivative of a function which is expressed as a fraction. The general formula for the quotient rule is:

Q' (gf' - g'f) / g2

where Q' is the derivative of the quotient, g is the denominator, and f is the numerator.

Application of the Quotient Rule to the Given Function

Let's start with the given function: f(x) 3x-5 / 5x2.

Step 1: Identifying the Numerator and Denominator

In this case, the numerator is f(x) 3x-5 and the denominator is g(x) 5x2.

Step 2: Find the Derivatives of the Numerator and Denominator

We need to differentiate both the numerator and the denominator:

Numerator: d/dx [3x-5] -15x-6 Denominator: d/dx [5x2] 1

Step 3: Apply the Quotient Rule

Using the quotient rule, we substitute the derivatives and the original functions into the formula:

After simplifying, we get:

Therefore, the derivative of the function is:

Method 2: Simplifying the Process with Substitution

As an alternative, we can simplify the process by using the properties of exponents to rewrite the function before applying the quotient rule:

f(x) (3x-5 / 5x2) (3/5) (x-5 / x2) (3/5) x-7

The derivative of this function can be found using the power rule:

It is noted that the two methods yield equivalent results, confirming the accuracy of our calculations.

Conclusion

In this guide, we have walked through the process of finding the derivative of the function 3x-5 / 5x2 using the quotient rule and an alternative method. Whether you are a student, a teacher, or someone interested in understanding the intricacies of calculus, this guide will serve as a useful reference. The quotient rule, while powerful, can be a bit complex, but with practice, it becomes an indispensable tool in your calculus toolkit.