Finding Third And Fourth Derivatives Of Y=x(x+1)/(x+2)(x+3)

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In the realm of calculus, derivatives play a pivotal role in understanding the rate at which a function's output changes with respect to its input. While the first derivative provides insights into the function's slope and rate of change, higher-order derivatives unveil more intricate aspects of its behavior, such as concavity and inflection points. In this comprehensive exploration, we delve into the process of finding the third and fourth derivatives of the function y = x(x+1)/(x+2)(x+3). This exercise not only demonstrates the application of differentiation rules but also sheds light on the significance of higher-order derivatives in mathematical analysis.

Understanding the Foundation: Differentiation Rules

Before embarking on the journey of finding the third and fourth derivatives, it's crucial to solidify our understanding of the fundamental differentiation rules that will serve as our guiding principles. These rules provide a systematic framework for calculating derivatives of various types of functions, including polynomials, rational functions, and composite functions. Mastering these rules is essential for navigating the complexities of differentiation and ensuring the accuracy of our results.

The Power Rule: A Cornerstone of Differentiation

The power rule stands as one of the most frequently employed and fundamental rules in differentiation. It elegantly states that the derivative of x^n, where n is any real number, is given by nx^(n-1). This rule serves as the bedrock for differentiating polynomial functions, which are formed by the sum of terms involving powers of x. For instance, the derivative of x^3 is 3x^2, and the derivative of x^(-2) is -2x^(-3). The power rule's simplicity and versatility make it an indispensable tool in the calculus arsenal.

The Quotient Rule: Navigating Rational Functions

When confronted with rational functions, which are expressed as the ratio of two functions, the quotient rule emerges as the guiding principle. This rule dictates that the derivative of u(x)/v(x) is given by (v(x)u'(x) - u(x)v'(x)) / (v(x))^2, where u'(x) and v'(x) represent the derivatives of u(x) and v(x), respectively. The quotient rule effectively breaks down the differentiation of a rational function into manageable steps, allowing us to systematically calculate its derivative. This rule is particularly relevant in our quest to find the derivatives of y = x(x+1)/(x+2)(x+3), as it involves a rational expression.

The Chain Rule: Unraveling Composite Functions

Composite functions, which involve a function nested within another function, require the application of the chain rule. This rule states that the derivative of f(g(x)) is given by f'(g(x)) * g'(x), where f'(g(x)) represents the derivative of the outer function f evaluated at the inner function g(x), and g'(x) represents the derivative of the inner function g(x). The chain rule effectively unravels the layers of a composite function, allowing us to differentiate it step by step. While the chain rule may not be directly applicable in our current problem, it remains a vital tool in the broader context of differentiation.

Preparing the Function: Simplifying the Expression

Before embarking on the process of differentiation, it's often advantageous to simplify the function as much as possible. This simplification can make the subsequent differentiation steps more manageable and reduce the likelihood of errors. In the case of y = x(x+1)/(x+2)(x+3), we can expand the numerator and denominator to obtain a more explicit expression.

Expanding the numerator, we have x(x+1) = x^2 + x. Similarly, expanding the denominator, we get (x+2)(x+3) = x^2 + 5x + 6. Therefore, the function can be rewritten as y = (x^2 + x) / (x^2 + 5x + 6). This simplified form will make the application of the quotient rule more straightforward.

The First Derivative: Unveiling the Rate of Change

Now that we have a solid foundation in differentiation rules and have simplified the function, we can embark on the journey of finding the derivatives. The first derivative, denoted as y', provides insights into the function's rate of change at any given point. To find the first derivative of y = (x^2 + x) / (x^2 + 5x + 6), we will employ the quotient rule.

Applying the quotient rule, we have:

y' = [(x^2 + 5x + 6)(2x + 1) - (x^2 + x)(2x + 5)] / (x^2 + 5x + 6)^2

Expanding the numerator and simplifying, we get:

y' = (2x^3 + 11x^2 + 17x + 6 - 2x^3 - 7x^2 - 5x) / (x^2 + 5x + 6)^2

y' = (4x^2 + 12x + 6) / (x^2 + 5x + 6)^2

Thus, the first derivative of y = x(x+1)/(x+2)(x+3) is y' = (4x^2 + 12x + 6) / (x^2 + 5x + 6)^2. This expression reveals the instantaneous rate of change of the function at any given value of x.

The Second Derivative: Unveiling Concavity

The second derivative, denoted as y'', takes us a step further in understanding the function's behavior. It provides information about the concavity of the function, indicating whether the graph is curving upwards (concave up) or downwards (concave down). To find the second derivative, we need to differentiate the first derivative, y' = (4x^2 + 12x + 6) / (x^2 + 5x + 6)^2, with respect to x. This again involves the application of the quotient rule.

Applying the quotient rule, we have:

y'' = {[(x^2 + 5x + 6)^2(8x + 12) - (4x^2 + 12x + 6) * 2(x^2 + 5x + 6)(2x + 5)] / (x^2 + 5x + 6)^4}

Simplifying the expression by factoring out (x^2 + 5x + 6) from the numerator, we get:

y'' = [(x^2 + 5x + 6)(8x + 12) - 2(4x^2 + 12x + 6)(2x + 5)] / (x^2 + 5x + 6)^3

Expanding and simplifying the numerator further, we obtain:

y'' = (-8x^3 - 60x^2 - 132x - 96) / (x^2 + 5x + 6)^3

Thus, the second derivative of y = x(x+1)/(x+2)(x+3) is y'' = (-8x^3 - 60x^2 - 132x - 96) / (x^2 + 5x + 6)^3. This expression provides insights into the concavity of the function at different points.

The Third Derivative: Unveiling the Rate of Change of Concavity

The third derivative, denoted as y''', delves even deeper into the function's behavior, revealing the rate at which the concavity is changing. It provides information about the inflection points of the function, where the concavity changes from upwards to downwards or vice versa. To find the third derivative, we need to differentiate the second derivative, y'' = (-8x^3 - 60x^2 - 132x - 96) / (x^2 + 5x + 6)^3, with respect to x. This step once again involves the application of the quotient rule.

Applying the quotient rule, we have:

y''' = {[(x^2 + 5x + 6)3(-24x2 - 120x - 132) - (-8x^3 - 60x^2 - 132x - 96) * 3(x^2 + 5x + 6)^2(2x + 5)] / (x^2 + 5x + 6)^6}

Simplifying the expression by factoring out (x^2 + 5x + 6)^2 from the numerator, we get:

y''' = [(x^2 + 5x + 6)(-24x^2 - 120x - 132) - 3(-8x^3 - 60x^2 - 132x - 96)(2x + 5)] / (x^2 + 5x + 6)^4

Expanding and simplifying the numerator further, we obtain:

y''' = (48x^4 + 480x^3 + 1680x^2 + 2688x + 1584) / (x^2 + 5x + 6)^4

Thus, the third derivative of y = x(x+1)/(x+2)(x+3) is y''' = (48x^4 + 480x^3 + 1680x^2 + 2688x + 1584) / (x^2 + 5x + 6)^4. This expression provides insights into the rate of change of concavity and helps identify inflection points.

The Fourth Derivative: Unveiling the Rate of Change of the Rate of Change of Concavity

The fourth derivative, denoted as y^(4), takes us to the highest level of derivative analysis in this exploration. It represents the rate of change of the rate of change of concavity, providing even more nuanced information about the function's behavior. To find the fourth derivative, we need to differentiate the third derivative, y''' = (48x^4 + 480x^3 + 1680x^2 + 2688x + 1584) / (x^2 + 5x + 6)^4, with respect to x. This final step involves yet another application of the quotient rule.

Applying the quotient rule, we have:

y^(4) = {[(x^2 + 5x + 6)4(192x3 + 1440x^2 + 3360x + 2688) - (48x^4 + 480x^3 + 1680x^2 + 2688x + 1584) * 4(x^2 + 5x + 6)^3(2x + 5)] / (x^2 + 5x + 6)^8}

Simplifying the expression by factoring out (x^2 + 5x + 6)^3 from the numerator, we get:

y^(4) = [(x^2 + 5x + 6)(192x^3 + 1440x^2 + 3360x + 2688) - 4(48x^4 + 480x^3 + 1680x^2 + 2688x + 1584)(2x + 5)] / (x^2 + 5x + 6)^5

Expanding and simplifying the numerator further, we obtain:

y^(4) = (-384x^5 - 4800x^4 - 22080x^3 - 50688x^2 - 59904x - 28512) / (x^2 + 5x + 6)^5

Thus, the fourth derivative of y = x(x+1)/(x+2)(x+3) is y^(4) = (-384x^5 - 4800x^4 - 22080x^3 - 50688x^2 - 59904x - 28512) / (x^2 + 5x + 6)^5. This expression provides the most detailed insights into the function's behavior, capturing the rate of change of the rate of change of concavity.

Conclusion: A Journey Through Higher-Order Derivatives

In this comprehensive exploration, we have successfully navigated the process of finding the third and fourth derivatives of the function y = x(x+1)/(x+2)(x+3). This journey has not only demonstrated the application of differentiation rules but has also highlighted the significance of higher-order derivatives in mathematical analysis. The first derivative provides insights into the function's rate of change, the second derivative reveals its concavity, the third derivative unveils the rate of change of concavity, and the fourth derivative captures the rate of change of the rate of change of concavity.

By understanding and applying these concepts, we gain a deeper appreciation for the intricate behavior of functions and their applications in various fields of mathematics, science, and engineering. The exploration of higher-order derivatives opens doors to a more nuanced understanding of mathematical relationships and their impact on the world around us.