Derivative Of Y = T(ln 11t)^2 A Step-by-Step Guide

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Introduction

In calculus, finding the derivative of a function is a fundamental operation. The derivative represents the instantaneous rate of change of a function with respect to its variable. In this article, we will delve into the process of finding the derivative of the function y = t(ln 11t)^2 with respect to t. This problem combines the product rule, chain rule, and the derivative of the natural logarithm function, providing a comprehensive exercise in differential calculus.

Understanding the Problem

Before we jump into the solution, let's break down the given function: y = t(ln 11t)^2. Here, y is a function of t, and we are tasked with finding dy/dt, which represents the derivative of y with respect to t. The function is a product of two terms: t and (ln 11t)^2. This immediately suggests that we will need to use the product rule. Additionally, the term (ln 11t)^2 involves a composite function, meaning we will also need to apply the chain rule.

Key Concepts to Remember

  1. Product Rule: The product rule states that if we have a function y = u(t)v(t), then its derivative dy/dt is given by:

    dy/dt = u'(t)v(t) + u(t)v'(t)

    where u'(t) and v'(t) are the derivatives of u(t) and v(t) with respect to t, respectively.

  2. Chain Rule: The chain rule is used to find the derivative of a composite function. If we have a function y = f(g(t)), then its derivative dy/dt is given by:

    dy/dt = f'(g(t)) * g'(t)

    where f'(g(t)) is the derivative of the outer function evaluated at the inner function, and g'(t) is the derivative of the inner function with respect to t.

  3. Derivative of Natural Logarithm: The derivative of the natural logarithm function ln(t) with respect to t is:

    d/dt [ln(t)] = 1/t

    More generally, if we have ln(at), where a is a constant, the derivative is:

    d/dt [ln(at)] = (1/at) * a = 1/t

Applying the Product Rule

Let's apply the product rule to our function y = t(ln 11t)^2. We can identify u(t) = t and v(t) = (ln 11t)^2. We need to find u'(t) and v'(t).

Finding u'(t)

The derivative of u(t) = t with respect to t is straightforward:

u'(t) = d/dt [t] = 1

Finding v'(t)

To find the derivative of v(t) = (ln 11t)^2, we need to apply the chain rule. Let's consider v(t) = [g(t)]^2, where g(t) = ln 11t. The chain rule tells us that:

v'(t) = 2[g(t)] * g'(t) = 2(ln 11t) * d/dt [ln 11t]

Now, we need to find the derivative of ln 11t with respect to t. As mentioned earlier, the derivative of ln(at) is 1/t, so:

d/dt [ln 11t] = 1/t

Therefore, v'(t) becomes:

v'(t) = 2(ln 11t) * (1/t) = (2 ln 11t) / t

Combining the Results Using the Product Rule

Now that we have u'(t) = 1 and v'(t) = (2 ln 11t) / t, we can apply the product rule:

dy/dt = u'(t)v(t) + u(t)v'(t)

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

dy/dt = (ln 11t)^2 + 2 ln 11t

Simplifying the Derivative

We can factor out ln 11t from the expression to simplify it further:

dy/dt = (ln 11t) * (ln 11t + 2)

Thus, the derivative of y = t(ln 11t)^2 with respect to t is:

dy/dt = (ln 11t) (ln 11t + 2)

Detailed Step-by-Step Solution

To recap, here's a detailed step-by-step solution:

  1. Identify the functions: Recognize y = t(ln 11t)^2 as a product of two functions, u(t) = t and v(t) = (ln 11t)^2.
  2. Apply the product rule: dy/dt = u'(t)v(t) + u(t)v'(t).
  3. Find u'(t): The derivative of u(t) = t is u'(t) = 1.
  4. Find v'(t): Apply the chain rule to v(t) = (ln 11t)^2. Let g(t) = ln 11t, so v(t) = [g(t)]^2. Then, v'(t) = 2[g(t)] * g'(t).
  5. Find g'(t): The derivative of g(t) = ln 11t is g'(t) = 1/t.
  6. Substitute g'(t) into v'(t): v'(t) = 2(ln 11t) * (1/t) = (2 ln 11t) / t.
  7. Substitute u'(t) and v'(t) into the product rule: dy/dt = 1 * (ln 11t)^2 + t * [(2 ln 11t) / t].
  8. Simplify: dy/dt = (ln 11t)^2 + 2 ln 11t.
  9. Factor out common terms: dy/dt = (ln 11t) * (ln 11t + 2).

Common Mistakes to Avoid

When finding derivatives, it’s easy to make mistakes. Here are a few common pitfalls to watch out for:

  1. Forgetting the Chain Rule: When dealing with composite functions like (ln 11t)^2, it's crucial to apply the chain rule correctly. A common mistake is to differentiate the outer function but forget to multiply by the derivative of the inner function.
  2. Incorrectly Applying the Product Rule: The product rule requires careful application. Make sure to differentiate each part correctly and combine them in the right order.
  3. Misunderstanding Logarithm Derivatives: The derivative of ln(at) is 1/t, not 1/(at). Always remember to account for the constant inside the logarithm.
  4. Algebraic Errors: Simplifying the derivative often involves algebraic manipulation. Double-check each step to avoid mistakes in factoring or combining terms.

Practical Applications of Derivatives

Derivatives are not just abstract mathematical concepts; they have numerous practical applications in various fields. Understanding derivatives is essential in physics, engineering, economics, and computer science.

Physics

In physics, derivatives are used to describe motion. For example:

  • Velocity: The derivative of an object’s position with respect to time gives its velocity.
  • Acceleration: The derivative of an object’s velocity with respect to time gives its acceleration.

Derivatives help physicists model and understand how objects move and interact.

Engineering

Engineers use derivatives to optimize designs and systems. For example:

  • Optimization Problems: Derivatives can be used to find the maximum or minimum values of functions, which is crucial in designing efficient structures and systems.
  • Control Systems: Derivatives are used in control systems to predict and control the behavior of dynamic systems.

Economics

In economics, derivatives are used to analyze marginal changes. For example:

  • Marginal Cost: The derivative of the cost function with respect to quantity gives the marginal cost, which is the cost of producing one additional unit.
  • Marginal Revenue: The derivative of the revenue function with respect to quantity gives the marginal revenue, which is the revenue generated by selling one additional unit.

Computer Science

Derivatives are used in machine learning and optimization algorithms. For example:

  • Gradient Descent: This optimization algorithm uses derivatives to find the minimum of a function, which is essential in training machine learning models.
  • Neural Networks: Derivatives are used to calculate the gradients needed to update the weights in a neural network during training.

Conclusion

Finding the derivative of y = t(ln 11t)^2 with respect to t involves a combination of the product rule, chain rule, and the derivative of the natural logarithm. By carefully applying these rules and simplifying the result, we found that dy/dt = (ln 11t)(ln 11t + 2). This exercise underscores the importance of understanding and correctly applying the fundamental rules of differential calculus. Derivatives are a powerful tool with wide-ranging applications, making their mastery essential for anyone studying mathematics, science, or engineering. By understanding the step-by-step process and avoiding common mistakes, you can confidently tackle similar problems and appreciate the practical significance of derivatives in various fields.

This comprehensive guide not only walks you through the solution but also reinforces the underlying concepts, common pitfalls, and real-world applications, ensuring a thorough understanding of the topic.