Introduction

Finding the nth derivative of complex functions is a common task in higher mathematics, and it often involves the application of various differentiation techniques such as the chain rule and the quotient rule. In this article, we will explore how to derive the nth derivative of the function

fx  ln(6x^2 - x - 1)

. We will walk through the process step-by-step and highlight the importance of identifying patterns and using recursion for more practical computations.

Step 1: First Derivative

To find the first derivative, we start by applying the chain rule. The derivative of the natural logarithm function is:

frac{d}{dx} ln(u) frac{1}{u} frac{du}{dx}

In our case, the inner function is

u  6x^2 - x - 1

. Therefore, we have:

[f'(x) frac{1}{6x^2 - x - 1} cdot frac{d}{dx}(6x^2 - x - 1)]

Computing the derivative of the inner function, we get:

[frac{d}{dx}(6x^2 - x - 1) 12x - 1]

Putting it all together, the first derivative is:

[f'(x) frac{12x - 1}{6x^2 - x - 1}]

Step 2: Higher Derivatives Using the Quotient Rule

To find the nth derivative, we use the quotient rule repeatedly. The derivative of a quotient is given by:

frac{d}{dx} left(frac{u}{v}right) frac{u'v - uv'}{v^2}

For our function, set u 12x - 1 and v 6x^2 - x - 1. Applying the quotient rule to find the second derivative, we have:

[f''(x) frac{(12)(6x^2 - x - 1) - (12x - 1)(12x - 1)}{(6x^2 - x - 1)^2}]

This may seem complex, but it's just the application of the formula. To simplify, we expand and combine like terms:

[f''(x) frac{72x^2 - 12x - 12 - (144x^2 - 24x 1)}{(6x^2 - x - 1)^2}]

Simplifying further, we get:

[f''(x) frac{72x^2 - 12x - 12 - 144x^2 24x - 1}{(6x^2 - x - 1)^2}]

[f''(x) frac{-72x^2 12x - 13}{(6x^2 - x - 1)^2}]

General Formula for Higher Derivatives

The general formula for the nth derivative of a quotient can be quite intricate, but often identifying patterns is key. To find the nth derivative, we can use the recursion formula derived from the quotient rule. Instead of manually computing each derivative, we can look for patterns and use the relationship between successive derivatives.

Example: Second Derivative

To find the second derivative, we can use the quotient rule again. Let's simplify the process by factoring the polynomial first:

6x^2 - x - 1 (3x - 1)(2x 1)

Now, we rewrite the function as:

[f(x) ln(3x - 1) ln(2x 1)]

Using the properties of logarithms, we can compute the first derivative of each term separately. The derivatives are:

[f'(x) frac{3}{3x - 1} frac{2}{2x 1}]

Following the same process for the second derivative, we get a pattern that can be generalized. Recognizing this pattern can help us find the nth derivative with greater efficiency.

Conclusion

For practical purposes, if you need the nth derivative for specific values of n, computing the derivatives step by step until you find a repeating pattern or develop a recursive relationship can be very helpful. If you need further assistance with the nth derivative or any other specific derivatives, feel free to ask! In this article, we have explored the process of finding the nth derivative of the function fx ln(6x^2 - x - 1) using the chain rule, quotient rule, and factoring. By recognizing patterns and using recursive relationships, we can simplify the process and make it more manageable. If you're facing a similar problem, try breaking it down into simpler parts and look for patterns to help you find the solution.