91Ó°ÊÓ

A derivative is a mathematical concept that represents the rate at which a function changes at any point. It provides a way to analyze how a function's output varies as the input changes by an infinitesimally small amount. In this exercise, we are interested in the derivative of the function \( f(x) = \frac{1}{(1+x)^2} \), which tells us how quickly \( f(x) \) is changing around a specific point.

• The derivative can be thought of as the slope of the tangent line to the function at a given point.
• It is symbolized as \( f'(x) \) or \( \frac{df}{dx} \), indicating that it is a function of the input variable \( x \).

By finding the derivative at \( a = 0 \), we are determining the slope of the tangent line to \( f(x) \) at this point, which helps us create the linear approximation. This approximation, \( L(x) = 1 - 2x \), is an excellent way to estimate function values close to \( x = 0 \).

The chain rule is an essential technique in differentiation, especially when dealing with composite functions like \( f(x) = \frac{1}{(1+x)^2} \). It allows us to find the derivative of a function composed of two or more simpler functions.
To use the chain rule, we identify two functions, an "outer" function and an "inner" function. Here:

• The outer function is \( u^{-2} \) where \( u = 1+x \).
• The inner function is \( 1+x \).

Using the chain rule involves differentiating the outer function with respect to the inner function, and then multiplying by the derivative of the inner function:\[ \frac{d}{dx} (1+x)^{-2} = -2(1+x)^{-3} \times 1 = -2(1+x)^{-3} \]This step is crucial for determining \( f'(x) \), enabling us to find the linear approximation.

Differentiation is the process of finding the derivative of a function. It is a fundamental operation in calculus used to study how functions change. The linear approximation technique requires differentiation to approximate function values near a specific point, helping us develop an understanding of more complex behaviors through simpler linear models.
In this exercise, we differentiated the function \( f(x) = \frac{1}{(1+x)^2} \) by applying rules of differentiation such as:

• Power Rule: Used to handle the form \( (1+x)^{-2} \).
• Chain Rule: As previously explained, applies to composite functions.

These rules work together to give us the derivative, \( f'(x) = -2(1+x)^{-3} \). By evaluating this derivative at \( a = 0 \), we can effectively apply it in the linear approximation formula, ultimately achieving a simpler expression that estimates the behavior of the original function around the given point.