Derivative Of Y = (8 + Ln T) / (3 - Ln T) With Respect To T

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Introduction

In the realm of calculus, finding the derivative of a function is a fundamental operation. The derivative, often denoted as dy/dt, represents the instantaneous rate of change of a function y with respect to a variable t. In simpler terms, it tells us how much y changes for a tiny change in t. This concept has vast applications across various fields, including physics, engineering, economics, and computer science.

This article delves into the process of finding the derivative of a specific function: y = (8 + ln t) / (3 - ln t). This function involves the natural logarithm (ln t), which adds a layer of complexity. To tackle this problem, we'll employ the quotient rule, a powerful tool in calculus for differentiating functions that are expressed as fractions.

The quotient rule is particularly useful when dealing with functions like the one we have, where both the numerator and the denominator contain variables. Mastering this rule is crucial for anyone venturing into calculus and its applications. We will meticulously break down each step, ensuring a clear and comprehensive understanding of the process. By the end of this article, you'll not only know how to find the derivative of this specific function but also gain a deeper appreciation for the quotient rule and its significance in calculus.

So, let's embark on this mathematical journey and unravel the intricacies of finding the derivative of y with respect to t.

Understanding the Quotient Rule

The quotient rule is a cornerstone of differential calculus, providing a systematic method for finding the derivative of a function that is expressed as the ratio of two other functions. In mathematical terms, if we have a function y that can be written as y = u(t) / v(t), where u(t) and v(t) are differentiable functions of t, then the derivative of y with respect to t, denoted as dy/dt, is given by the following formula:

dy/dt = [v(t) * (du/dt) - u(t) * (dv/dt)] / [v(t)]^2

Let's break down this formula to understand its components:

  • u(t): This represents the function in the numerator of the original expression. In our case, u(t) = 8 + ln t.
  • v(t): This represents the function in the denominator of the original expression. In our case, v(t) = 3 - ln t.
  • du/dt: This is the derivative of u(t) with respect to t. It represents the rate of change of the numerator function.
  • dv/dt: This is the derivative of v(t) with respect to t. It represents the rate of change of the denominator function.
  • [v(t)]^2: This is the square of the denominator function.

The quotient rule essentially states that the derivative of a quotient is equal to the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator. This might sound complex, but with practice, it becomes a straightforward process.

The importance of the quotient rule stems from its ability to handle a wide range of functions that are expressed as fractions. Many mathematical models in science and engineering involve ratios of functions, making the quotient rule an indispensable tool for analyzing their behavior. Understanding and applying this rule correctly is crucial for mastering calculus and its applications.

Applying the Quotient Rule to y = (8 + ln t) / (3 - ln t)

Now, let's apply the quotient rule to our specific function, y = (8 + ln t) / (3 - ln t). This will involve identifying the numerator and denominator functions, finding their derivatives, and then plugging them into the quotient rule formula. This step-by-step process will not only give us the final answer but also solidify our understanding of the quotient rule.

1. Identify u(t) and v(t):

As we discussed earlier, u(t) represents the numerator and v(t) represents the denominator. In our case:

  • u(t) = 8 + ln t
  • v(t) = 3 - ln t

2. Find du/dt and dv/dt:

Next, we need to find the derivatives of u(t) and v(t) with respect to t. Recall that the derivative of a constant is zero, and the derivative of ln t is 1/t.

  • du/dt = d(8 + ln t)/dt = 0 + (1/t) = 1/t
  • dv/dt = d(3 - ln t)/dt = 0 - (1/t) = -1/t

3. Apply the Quotient Rule Formula:

Now that we have u(t), v(t), du/dt, and dv/dt, we can plug them into the quotient rule formula:

dy/dt = [v(t) * (du/dt) - u(t) * (dv/dt)] / [v(t)]^2

Substituting the values, we get:

dy/dt = [(3 - ln t) * (1/t) - (8 + ln t) * (-1/t)] / (3 - ln t)^2

This is the initial application of the quotient rule. In the next section, we will simplify this expression to obtain the final derivative.

Simplifying the Expression

After applying the quotient rule, we arrived at the following expression for dy/dt:

dy/dt = [(3 - ln t) * (1/t) - (8 + ln t) * (-1/t)] / (3 - ln t)^2

Now, our task is to simplify this expression to obtain the final derivative in its most concise form. This involves algebraic manipulation, such as distributing terms, combining like terms, and simplifying fractions. Let's break down the simplification process step by step.

1. Distribute the terms in the numerator:

First, we distribute the 1/t and -1/t terms in the numerator:

dy/dt = [(3/t) - (ln t)/t + (8/t) + (ln t)/t] / (3 - ln t)^2

2. Combine like terms in the numerator:

Next, we combine the like terms in the numerator. Notice that -(ln t)/t and +(ln t)/t cancel each other out:

dy/dt = [(3/t) + (8/t)] / (3 - ln t)^2

dy/dt = (11/t) / (3 - ln t)^2

3. Simplify the complex fraction:

Finally, we simplify the complex fraction by dividing (11/t) by (3 - ln t)^2. This is equivalent to multiplying (11/t) by the reciprocal of (3 - ln t)^2, which is 1 / (3 - ln t)^2:

dy/dt = (11/t) * [1 / (3 - ln t)^2]

dy/dt = 11 / [t * (3 - ln t)^2]

This is the simplified form of the derivative. It represents the instantaneous rate of change of y with respect to t for the given function. This final expression is much cleaner and easier to work with than the initial expression we obtained after applying the quotient rule.

Final Answer and Conclusion

After meticulously applying the quotient rule and simplifying the resulting expression, we have successfully found the derivative of y = (8 + ln t) / (3 - ln t) with respect to t. The final answer is:

dy/dt = 11 / [t * (3 - ln t)^2]

This result tells us how the value of y changes as t changes. It's a crucial piece of information for understanding the behavior of the function and its applications in various fields.

In conclusion, finding the derivative of a function using the quotient rule involves a systematic approach: identifying the numerator and denominator functions, finding their individual derivatives, applying the quotient rule formula, and simplifying the resulting expression. This process, while initially appearing complex, becomes more manageable with practice and a solid understanding of the underlying concepts.

The quotient rule is a powerful tool in calculus, enabling us to differentiate a wide range of functions that are expressed as fractions. Its applications extend far beyond theoretical mathematics, playing a vital role in solving real-world problems in physics, engineering, economics, and other disciplines. Mastering the quotient rule is an essential step for anyone seeking to delve deeper into the world of calculus and its applications.

We hope this article has provided a clear and comprehensive explanation of how to find the derivative of y = (8 + ln t) / (3 - ln t) using the quotient rule. By understanding the steps involved and practicing similar problems, you can confidently tackle more complex differentiation challenges in the future.