Derivative of the Function y ln(1 - 3x): Applying the Chain Rule

Understanding the derivative of a function like y ln(1 - 3x) is crucial in calculus, especially when dealing with functions within logarithms. When approaching such problems, the chain rule is a powerful tool that simplifies the process. This article will provide a detailed explanation of how to find the derivative of y ln(1 - 3x), step-by-step, using the chain rule and other derivative rules.

Understanding the Chain Rule

The chain rule is a fundamental concept in calculus for differentiating composite functions. It states that if y is a differentiable function of u, and u is a differentiable function of x, then the composite function y f(g(x)) is differentiable, and its derivative is given by:

(frac{dy}{dx} frac{dy}{du} cdot frac{du}{dx})

Deriving the Function y ln(1 - 3x)

To find the derivative of y ln(1 - 3x), we can break it down using the chain rule. Here's how:

Step 1: Identify the Inside and Outer Functions

Let's rewrite the function as y ln(u), where u 1 - 3x. This identifies ln(u) as the outer function and u 1 - 3x as the inner function.

Step 2: Differentiate the Outer Function

The derivative of ln(u) with respect to u is:

(frac{dy}{du} frac{1}{u})

Step 3: Differentiate the Inner Function

The derivative of the inner function u 1 - 3x with respect to x is:

(frac{du}{dx} -3)

Step 4: Apply the Chain Rule

Now, applying the chain rule:

(frac{dy}{dx} frac{dy}{du} cdot frac{du}{dx} frac{1}{u} cdot (-3) -frac{3}{u})

Substituting back the inner function u 1 - 3x, we get:

(frac{dy}{dx} -frac{3}{1 - 3x})

Explanation and Example

Let's illustrate this process further with the given function y ln(1 - 3x) step-by-step:

Step 1: Define the Inner Function

u 1 - 3x

Step 2: Derivative of the Outer Function

(frac{d}{du}(ln(u)) frac{1}{u})

Step 3: Derivative of the Inner Function

(frac{du}{dx} -3)

Step 4: Apply the Chain Rule

Combining these steps:

(frac{dy}{dx} frac{1}{u} cdot (-3) -frac{3}{1 - 3x})

Additional Practice and Resources

Practicing similar problems can significantly enhance your understanding of the chain rule and logarithmic differentiation. Here are a few more tips to help you master these concepts:

Practice differentiating a variety of logarithmic functions. Use online resources and videos to supplement your learning. Work through examples and exercises in textbooks and problem sets.

By familiarizing yourself with the chain rule and practicing regularly, you can confidently tackle more complex calculus problems involving logarithmic functions.