Derivative Of Y = Ln(√((3 - 8x^3) / (18x^3 - 12x))) A Step-by-Step Solution

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In this article, we will walk through the process of finding the derivative of a complex logarithmic function. Specifically, we will tackle the following expression:

y = ln(√((3 - 8x^3) / (18x^3 - 12x)))

This problem requires a combination of logarithmic properties, the chain rule, and the quotient rule. By breaking it down step-by-step, we can clearly illustrate how to arrive at the solution.

Understanding the Problem and Key Concepts

Before we dive into the calculations, it's essential to understand the core concepts involved. Our main goal is to find dy/dx, which represents the derivative of y with respect to x. This tells us how y changes as x changes. The given function, y, involves a natural logarithm (ln) and a square root, which contains a rational function. Therefore, we'll need to utilize several key rules of calculus:

  1. Logarithmic Properties: These properties help simplify complex logarithmic expressions. The most relevant ones for this problem are:
    • ln(a/b) = ln(a) - ln(b)
    • ln(a^n) = n * ln(a)
  2. Chain Rule: This rule is used to differentiate composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
  3. Quotient Rule: This rule is used to differentiate functions that are expressed as a quotient. If y = u(x) / v(x), then dy/dx = (v(x)u'(x) - u(x)v'(x)) / (v(x))^2.

With these concepts in mind, let's proceed to solve the problem.

Step 1: Simplifying the Expression Using Logarithmic Properties

The initial expression looks quite complex, but we can simplify it significantly using logarithmic properties. First, we recognize that the square root can be expressed as a power of 1/2.

y = ln(((3 - 8x^3) / (18x^3 - 12x))^(1/2))

Now, we can use the property ln(a^n) = n * ln(a) to bring the exponent outside the logarithm:

y = (1/2) * ln((3 - 8x^3) / (18x^3 - 12x))

Next, we can apply the property ln(a/b) = ln(a) - ln(b) to separate the fraction inside the logarithm:

y = (1/2) * [ln(3 - 8x^3) - ln(18x^3 - 12x)]

This simplified form is much easier to differentiate. Now, we can move on to the next step, which involves applying the chain rule.

Step 2: Applying the Chain Rule and Differentiation

Now that we have simplified the expression, we can differentiate it with respect to x. Remember, we have:

y = (1/2) * [ln(3 - 8x^3) - ln(18x^3 - 12x)]

We will apply the chain rule to each logarithmic term. The derivative of ln(u) with respect to x is (1/u) * du/dx. Let's differentiate each term separately.

Differentiating the First Term: ln(3 - 8x^3)

Let u = 3 - 8x^3. Then, du/dx = -24x^2. Applying the chain rule:

d/dx [ln(3 - 8x^3)] = (1 / (3 - 8x^3)) * (-24x^2) = -24x^2 / (3 - 8x^3)

Differentiating the Second Term: ln(18x^3 - 12x)

Let v = 18x^3 - 12x. Then, dv/dx = 54x^2 - 12. Applying the chain rule:

d/dx [ln(18x^3 - 12x)] = (1 / (18x^3 - 12x)) * (54x^2 - 12) = (54x^2 - 12) / (18x^3 - 12x)

Now, we combine these results and remember the (1/2) factor from the original expression:

dy/dx = (1/2) * [(-24x^2 / (3 - 8x^3)) - ((54x^2 - 12) / (18x^3 - 12x))]

Step 3: Simplifying the Derivative

We now have the derivative, but it looks quite complex. We need to simplify it to obtain a more manageable expression. Let's focus on the expression inside the brackets:

(-24x^2 / (3 - 8x^3)) - ((54x^2 - 12) / (18x^3 - 12x))

To combine these fractions, we need a common denominator. Notice that we can factor out a 6x from the denominator of the second term:

18x^3 - 12x = 6x(3x^2 - 2)

So, the expression becomes:

(-24x^2 / (3 - 8x^3)) - ((54x^2 - 12) / (6x(3x^2 - 2)))

Now, let's find a common denominator and combine the fractions. This step can be quite tedious, but it's essential to arrive at the simplest form.

First, factor out constants where possible:

(-24x^2 / (3 - 8x^3)) - (6(9x^2 - 2) / (6x(3x^2 - 2)))

Simplify the second term by canceling out the 6:

(-24x^2 / (3 - 8x^3)) - ((9x^2 - 2) / (x(3x^2 - 2)))

The common denominator will be (3 - 8x^3) * x(3x^2 - 2). Multiply each fraction by the appropriate factors to get the common denominator:

[(-24x^2 * x(3x^2 - 2)) - ((9x^2 - 2) * (3 - 8x^3))] / [(3 - 8x^3) * x(3x^2 - 2)]

Expand the numerator:

(-72x^5 + 48x^3) - (27x^2 - 72x^5 - 6 + 16x^3)

Combine like terms in the numerator:

-72x^5 + 48x^3 - 27x^2 + 72x^5 + 6 - 16x^3

32x^3 - 27x^2 + 6

So, the combined fraction is:

(32x^3 - 27x^2 + 6) / [(3 - 8x^3) * x(3x^2 - 2)]

Now, we multiply by the (1/2) factor from the original derivative:

dy/dx = (1/2) * [(32x^3 - 27x^2 + 6) / ((3 - 8x^3) * x(3x^2 - 2))]

dy/dx = (32x^3 - 27x^2 + 6) / [2x(3 - 8x^3)(3x^2 - 2)]

Step 4: Final Simplified Form

The derivative is now in a simplified form. However, we can attempt to factor the numerator and denominator further to see if any cancellations are possible. After careful inspection, it appears that there are no obvious factors to cancel.

Thus, the final simplified derivative is:

dy/dx = (32x^3 - 27x^2 + 6) / [2x(3 - 8x^3)(3x^2 - 2)]

This is the derivative of the given logarithmic function.

Summary of Steps

Let's recap the steps we took to find the derivative:

  1. Simplify the Expression: Used logarithmic properties to simplify the original function.
  2. Apply the Chain Rule: Differentiated the simplified function using the chain rule.
  3. Simplify the Derivative: Combined fractions and simplified the resulting expression.
  4. Final Simplified Form: Presented the derivative in its simplest form.

By following these steps, we successfully found the derivative of the given logarithmic function. This process demonstrates the importance of understanding logarithmic properties and the chain rule in calculus.

Conclusion

Finding the derivative of complex functions like this requires a solid understanding of calculus principles and algebraic manipulation. The combination of logarithmic properties, the chain rule, and the quotient rule allows us to break down the problem into manageable steps. While the algebra can be tedious, the systematic approach ensures that we arrive at the correct solution. Remember to always simplify your expressions as much as possible to make the final result clear and concise. Through practice and careful attention to detail, you can master these techniques and confidently tackle similar problems in calculus.