91Ó°ÊÓ

The quotient rule is a key concept in calculus. It is used when you need to differentiate a function that is the ratio of two other functions. Recall that if you have a function like \( y = \frac{u}{v} \), where \( u \) and \( v \) are functions of \( x \), then the quotient rule states:

\[ y' = \frac{v(u') - u(v')}{v^2} \]

Here, \( u' \) and \( v' \) are the derivatives of \( u \) and \( v \), respectively. The numerator of the quotient rule formula involves the product of the denominator function \( v \) and the derivative of the numerator function \( u' \), minus the product of the numerator function \( u \) and the derivative of the denominator function \( v' \). The denominator of the quotient rule formula is simply the square of the denominator function \( v^2 \).

It's essential to identify the numerator and denominator functions and their respective derivatives correctly to apply the quotient rule accurately.

A derivative represents the rate of change of a function with respect to one of its variables, typically \( x \). When you differentiate a function, you are essentially finding how the function's value changes as its input changes. For this problem, our function \( y = \frac{x-1}{x+1} \) is a ratio of two simpler functions:

• Numerator: \( u = x-1 \)
• Denominator: \( v = x+1 \)

First, identify the derivatives of these simpler functions:

\( u' = 1 \)
\( v' = 1 \)

Now that you have these derivatives, you can utilize the quotient rule to differentiate the original function.

Simplification is a crucial step in differentiation to achieve a neat, manageable, and interpretable result. After applying the quotient rule to our function \( y = \frac{x-1}{x+1} \), we get:

\[ y' = \frac{(x+1)(1) - (x-1)(1)}{(x+1)^2} \]

The next steps involve simplifying this expression:

1. Expand the terms in the numerator: \( (x+1) - (x-1) \)
2. Combine like terms: \( x + 1 - x + 1 = 2 \)
3. Thus the numerator simplifies to \( 2 \).

The expression becomes:
\[ y' = \frac{2}{(x+1)^2} \]

So, the simplified derivative of \( y = \frac{x-1}{x+1} \) is \( \frac{2}{(x+1)^2} \). Simplifying correctly ensures clarity and helps in understanding the behavior of the function more effectively.