Problem 66 - Thomas Calculus

Question

In Exercises \(55-68,\) use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$ y=\sqrt{\frac{(x+1)^{10}}{(2 x+1)^{5}}} $$

Step-by-Step Solution

Answer

Differentiate using logarithmic differentiation; derivative is obtained by applying chain, product, and quotient rules.

Step 1: Take the Natural Log

\(y = \sqrt{\frac{(x+1)^{10}}{(2x+1)^5}} = \left(\frac{(x+1)^{10}}{(2x+1)^5}\right)^{1/2}\)
\(\ln y = \frac{1}{2}[10\ln(x+1) - 5\ln(2x+1)] = 5\ln(x+1) - \frac{5}{2}\ln(2x+1)\)

Step 2: Differentiate Both Sides

\(\frac{y'}{y} = \frac{5}{x+1} - \frac{5}{2} \cdot \frac{2}{2x+1} = \frac{5}{x+1} - \frac{5}{2x+1}\)

Step 3: Solve for y'

\(y' = y\left(\frac{5}{x+1} - \frac{5}{2x+1}\right) = \sqrt{\frac{(x+1)^{10}}{(2x+1)^5}} \cdot \frac{5(2x+1) - 5(x+1)}{(x+1)(2x+1)}\)
\(= \sqrt{\frac{(x+1)^{10}}{(2x+1)^5}} \cdot \frac{5x}{(x+1)(2x+1)}\)

Key Concepts

Calculus

Calculus is a branch of mathematics that allows us to study changes between values that are related by a function. There are two main concepts in calculus: differentiation and integration. Each serves a unique purpose. Differentiation focuses on finding the rate at which a quantity changes. It plays a central role in determining the behavior and properties of functions. Integration, on the other hand, is concerned with finding total quantities based on rates of change. We use differentiation extensively when we want to determine slopes, rates, and the motion of objects.

Logarithmic differentiation, which we use in this exercise, is a specific technique within the realm of differentiation. It simplifies the process of differentiating complex or cumbersome functions. By taking the natural logarithm of both sides of an equation, we transform the product, quotient, or power functions into a more manageable form. This technique is particularly helpful when the function involves variables raised to high powers or products/quotients of multiple functions.

Derivative

The derivative of a function is a fundamental concept in calculus that measures how a function changes as its input changes. It is a core idea for analyzing rates of change and slopes of functions. The notation for the derivative of a function \( y \) with respect to \( x \) is \( \frac{dy}{dx} \).

• When calculating derivatives, one often uses rules such as the product rule, quotient rule, and chain rule.
• In logarithmic differentiation, the power or exponentials in functions are addressed by first taking the natural log, turning multiplication and division into addition and subtraction, which simplifies differentiation. For example, log properties allow \( \ln(a^b) = b\ln(a) \).

For the given function, taking the natural log simplifies its differentiation. Differentiation then becomes a process of applying basic rules to expressions that are easier to handle, allowing us to find the derivative in a more straightforward way. This technique helps simplify complicated differentiation problems into ones using primarily addition, subtraction, and multiplication of known derivatives.

Advanced Mathematics

Advanced mathematics encompasses more than just techniques—it's about developing new strategies for solving problems that appear complex at first glance. In this exercise, we leverage the power of logarithms, which are part of higher-level mathematics, to solve a problem that might otherwise seem complicated. Logarithmic differentiation is a tool that showcases how various advanced concepts can be combined in calculus to tackle intricate problems efficiently.

• By utilizing the properties of logarithms, we turn multiplication into addition and division into subtraction, making it easier to work with.
• Understanding advanced mathematical concepts allows us to make strategic use of mathematical properties, enhancing our problem-solving toolkit.
• These strategies are essential for fields requiring precision and complex computations, such as engineering, physics, and computer science.

The technique of logarithmic differentiation not only simplifies the differentiation of complicated functions but also illustrates the beauty of mathematical ingenuity, where complex problems are deconstructed into more straightforward steps.