Evaluating The Derivative Of (x^(3/4) * Sqrt(x^2 + 1)) / (3x+2)^5
In this article, we delve into the process of evaluating the derivative of a complex function involving radicals, polynomials, and quotients. Specifically, we will tackle the problem of finding the derivative with respect to x of the function:
f(x) = \frac{x^{\frac{3}{4}} \sqrt{x^2 + 1}}{(3x+2)^5}
This function presents a challenge due to its intricate structure, which combines fractional exponents, square roots, and polynomial terms in both the numerator and the denominator. To effectively compute the derivative, we will employ a combination of techniques, including the quotient rule, product rule, and chain rule, along with the properties of exponents and radicals. This comprehensive approach will allow us to systematically break down the problem into manageable steps and arrive at the final derivative.
The function we aim to differentiate is a quintessential example of a complex expression that necessitates a multifaceted approach. It is not simply a matter of applying a single differentiation rule; rather, it requires a strategic combination of several rules. The quotient rule is essential for handling the overall fraction, while the product rule will be instrumental in differentiating the numerator, which itself consists of two functions multiplied together. Additionally, the chain rule will be indispensable for dealing with the composite functions, such as the square root and the power of the denominator. This intricate interplay of rules underscores the importance of a strong foundational understanding of calculus principles and the ability to recognize the appropriate techniques for different functional forms.
Moreover, the presence of fractional exponents and radicals adds another layer of complexity. These elements necessitate the application of exponent rules and the chain rule in a nuanced manner. The fractional exponent in the term x^(3/4) requires careful treatment when applying the power rule, while the square root in the term √(x^2 + 1) demands the use of the chain rule in conjunction with the power rule. By meticulously applying these rules and paying close attention to detail, we can navigate the intricacies of the function and arrive at the correct derivative. This exercise not only reinforces our understanding of differentiation techniques but also highlights the importance of precision and accuracy in mathematical manipulations.
To simplify the differentiation process, we will first apply logarithmic differentiation. This technique is particularly useful when dealing with functions that involve products, quotients, and exponents. Logarithmic differentiation allows us to transform a complex expression into a more manageable form by taking the natural logarithm of both sides of the equation. This transformation effectively converts products into sums, quotients into differences, and exponents into products, thereby simplifying the subsequent differentiation steps. The underlying principle behind logarithmic differentiation lies in the properties of logarithms, which enable us to manipulate complex expressions into simpler forms that are easier to differentiate.
Let's denote the function as:
y = \frac{x^{\frac{3}{4}} \sqrt{x^2 + 1}}{(3x+2)^5}
Taking the natural logarithm of both sides, we get:
ln(y) = ln \left( \frac{x^{\frac{3}{4}} \sqrt{x^2 + 1}}{(3x+2)^5} \right)
Using the properties of logarithms, we can rewrite the right-hand side as:
ln(y) = ln(x^{\frac{3}{4}}) + ln(\sqrt{x^2 + 1}) - ln((3x+2)^5)
Further simplifying using logarithm properties:
ln(y) = \frac{3}{4}ln(x) + \frac{1}{2}ln(x^2 + 1) - 5ln(3x+2)
This transformation is a crucial step in simplifying the differentiation process. By converting the original complex expression into a sum and difference of logarithmic terms, we have effectively linearized the problem. This linearization allows us to apply the differentiation rules more easily and systematically. The logarithmic function acts as a powerful tool in this context, transforming a complex multiplicative and exponential structure into a simpler additive structure. This simplification is particularly advantageous when dealing with functions that involve multiple factors, quotients, and exponents, as it reduces the risk of errors and makes the differentiation process more manageable.
Now, we differentiate both sides of the equation with respect to x. This step involves applying the chain rule to the left-hand side and the standard differentiation rules to the right-hand side. The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. In this context, the left-hand side, ln(y), is a composite function where the outer function is the natural logarithm and the inner function is y(x). Applying the chain rule, we differentiate the outer function with respect to the inner function and then multiply by the derivative of the inner function with respect to x. This process ensures that we account for the rate of change of the outer function as it is affected by the rate of change of the inner function.
\frac{d}{dx} ln(y) = \frac{d}{dx} \left( \frac{3}{4}ln(x) + \frac{1}{2}ln(x^2 + 1) - 5ln(3x+2) \right)
Applying the chain rule on the left side, we get:
\frac{1}{y} \frac{dy}{dx} = \frac{3}{4} \frac{d}{dx} ln(x) + \frac{1}{2} \frac{d}{dx} ln(x^2 + 1) - 5 \frac{d}{dx} ln(3x+2)
Now, we differentiate each term on the right-hand side:
\frac{1}{y} \frac{dy}{dx} = \frac{3}{4} \cdot \frac{1}{x} + \frac{1}{2} \cdot \frac{2x}{x^2 + 1} - 5 \cdot \frac{3}{3x+2}
Simplifying the equation:
\frac{1}{y} \frac{dy}{dx} = \frac{3}{4x} + \frac{x}{x^2 + 1} - \frac{15}{3x+2}
This step is crucial in isolating the derivative dy/dx. By differentiating the logarithmic form, we have transformed the original complex differentiation problem into a simpler algebraic equation. The derivatives of the logarithmic terms are straightforward to compute, and the resulting equation is linear in dy/dx. This linearity allows us to easily solve for dy/dx by multiplying both sides of the equation by y. The process of differentiating the logarithmic form and simplifying the resulting equation demonstrates the power of logarithmic differentiation as a technique for handling complex functions.
To find , we multiply both sides of the equation by y:
\frac{dy}{dx} = y \left( \frac{3}{4x} + \frac{x}{x^2 + 1} - \frac{15}{3x+2} \right)
Substitute the original expression for y:
\frac{dy}{dx} = \frac{x^{\frac{3}{4}} \sqrt{x^2 + 1}}{(3x+2)^5} \left( \frac{3}{4x} + \frac{x}{x^2 + 1} - \frac{15}{3x+2} \right)
This step is the culmination of the logarithmic differentiation process. By multiplying both sides of the equation by the original function y, we have successfully isolated the derivative dy/dx. The resulting expression represents the derivative of the original complex function. It is a product of the original function and a sum of simpler terms, each of which corresponds to the derivative of a logarithmic term. This final expression encapsulates the rate of change of the original function with respect to x. It provides a complete and accurate representation of the derivative, taking into account all the intricacies of the original function's structure.
The expression we have obtained is a general formula for the derivative of the original function. It can be used to evaluate the derivative at any point x in the domain of the function. However, it is important to note that the expression may not be in its simplest form. Further algebraic simplification may be possible, depending on the specific application or the desired level of detail. Nevertheless, the expression we have obtained is a valid and complete representation of the derivative, and it serves as a testament to the power and versatility of logarithmic differentiation as a technique for handling complex functions.
The derivative of the given function is:
\frac{dy}{dx} = \frac{x^{\frac{3}{4}} \sqrt{x^2 + 1}}{(3x+2)^5} \left( \frac{3}{4x} + \frac{x}{x^2 + 1} - \frac{15}{3x+2} \right)
This final expression represents the derivative of the original function, capturing its rate of change with respect to x. It is the result of a systematic application of logarithmic differentiation, which allowed us to break down the complex function into manageable components and apply the appropriate differentiation rules. The expression is a combination of the original function and a sum of simpler terms, each of which reflects the contribution of a specific component of the original function to the overall rate of change. This final result serves as a testament to the power and versatility of calculus techniques in handling complex mathematical expressions.
The process of obtaining this derivative has involved several key steps, including logarithmic transformation, differentiation of logarithmic terms, and algebraic manipulation. Each step has played a crucial role in simplifying the problem and arriving at the final solution. Logarithmic differentiation, in particular, has proven to be an invaluable tool in handling the complex structure of the original function. By transforming the function into a logarithmic form, we have effectively linearized the problem and made it amenable to the application of standard differentiation rules. The final result stands as a culmination of these efforts, providing a complete and accurate representation of the derivative of the original function.
In conclusion, evaluating the derivative of a complex function requires a combination of techniques and a systematic approach. Logarithmic differentiation, along with the chain rule, product rule, and quotient rule, provides a powerful framework for tackling such problems. The final result, while seemingly complex, is a precise representation of the function's rate of change and showcases the elegance and effectiveness of calculus in solving intricate mathematical challenges.
In this detailed walkthrough, we have successfully computed the derivative of a complex function using logarithmic differentiation. This method proves invaluable when dealing with functions involving products, quotients, and powers. By systematically applying the properties of logarithms and differentiation rules, we were able to transform the original problem into a more manageable form and arrive at the final derivative. This exercise highlights the importance of mastering various calculus techniques to tackle complex mathematical problems efficiently. The resulting derivative expression provides valuable insights into the function's behavior and rate of change, demonstrating the power and utility of calculus in mathematical analysis.
The application of logarithmic differentiation in this context underscores its effectiveness as a problem-solving tool. The ability to transform a complex multiplicative expression into a simpler additive form is a key advantage of this technique. By taking the natural logarithm of both sides of the equation, we were able to convert products into sums, quotients into differences, and powers into products. This transformation significantly simplified the differentiation process, allowing us to apply the standard differentiation rules more easily and accurately. The success of this approach highlights the importance of recognizing the appropriate techniques for different types of functions and problems.
Moreover, this exercise reinforces the importance of a strong foundation in calculus principles. The successful computation of the derivative required a thorough understanding of the chain rule, product rule, quotient rule, and the properties of logarithms. Each of these concepts played a crucial role in the solution process. The ability to apply these concepts correctly and systematically is essential for tackling complex calculus problems. This exercise serves as a valuable learning experience, demonstrating the interconnectedness of various calculus concepts and the importance of mastering them to solve challenging problems.
In summary, the evaluation of the derivative of the given complex function showcases the power and versatility of logarithmic differentiation. This technique, combined with a solid understanding of calculus principles, provides a robust framework for tackling complex mathematical problems. The resulting derivative expression offers valuable insights into the function's behavior and rate of change, highlighting the importance of calculus in mathematical analysis and its applications in various fields of science and engineering.
