91Ó°ÊÓ

The chain rule is a fundamental tool in calculus used for finding the derivative of composite functions. A composite function is a function composed of two or more functions, such as \( y = \bigg(\frac{x+1}{1-x}\bigg)^{2} \). In this example, the outer function is \( u^2 \) and the inner function is \( u = \frac{x+1}{1-x} \). To solve this using the chain rule, we first find the derivative of the outer function with respect to the inner function, then multiply it by the derivative of the inner function with respect to \( x \). This can be expressed as:

• Step 1: Let \( u = \frac{x+1}{1-x} \).
• Step 2: The outer function is \( y = u^2 \), so the derivative is \( \frac{dy}{du} = 2u \).
• Step 3: Find \( \frac{du}{dx} \) of the inner function using appropriate differentiation techniques.

By combining these steps, we obtain the overall derivative \( \frac{dy}{dx} = 2u \frac{du}{dx} \), resulting in \( \frac{4(x+1)}{(1-x)^3} \).

The quotient rule is used to differentiate functions that are the ratio of two other functions. If you have a function \( u = \frac{f(x)}{g(x)} \), then the derivative \( \frac{du}{dx} \) is found using:
o\frac{d}{dx}\bigg(\frac{f(x)}{g(x)}\bigg) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}.oFor our inner function \( u = \frac{x+1}{1-x} \), we identify:

• \( f(x) = x+1 \)
• \( g(x) = 1-x \)

1. Compute \( f'(x) = 1 \) and \( g'(x) = -1 \).
2. Using the quotient rule,
   
   \( \frac{du}{dx} = \frac{(1)(1-x) - (x+1)(-1)}{(1-x)^2} \)
3. Simplify to \( \frac{1 - x + x + 1}{(1-x)^2} = \frac{2}{(1-x)^2} \).
This provides the derivative of the inner function needed for applying the chain rule.

Differentiation techniques are methods used to find the derivative of functions. Some of the core techniques include:

• Product Rule: Used when differentiating the product of two functions.
• Quotient Rule: Essential for functions that are the ratio of two other functions, as seen in our example.
• Chain Rule: Crucial for composite functions to differentiate each part step by step.

To put it into context, let's revisit the derivative of \( y = \bigg(\frac{x+1}{1-x}\bigg)^{2} \):

1. Identify as a composite function, with the outer function \( u^2 \) and inner function \( \frac{x+1}{1-x} \).
2. Apply the chain rule: \( \frac{dy}{dx} = 2u \frac{du}{dx} \).
3. Use the quotient rule to find \( \frac{du}{dx} \).
4. Combine the results for the overall derivative.

By mastering these techniques, you can tackle a wide array of differentiation problems with ease.