91Ó°ÊÓ

Differentiation is a fundamental process in calculus that assists us in understanding how functions change. At its core, it is about finding the derivative, which represents the rate at which a function’s output changes with respect to a change in its input. For the function given in the exercise, expressing it as \( f(x) = \frac{x}{x+1} \), differentiation helps to find the slope of the tangent line at any point \( x \). This slope is crucial for linear approximation.

When we differentiate \( f(x) = \frac{x}{x+1} \) using the quotient rule, which is particularly suited for functions written as fractions, we simplify the problem of calculating change. It's important to remember to:

• Formulate a fraction as \( \frac{u}{v} \) where both \( u \) and \( v \) are functions of \( x \).
• Apply the quotient rule, which is \( \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \).

The result, in this case, \( f'(x) = \frac{1}{(x+1)^2} \), helps define the rate at which the function \( f(x) \) changes, which is vital for the linearization process.

The quotient rule is a technique in differentiation used to find the derivative of a quotient of two functions, often expressed as \( \frac{u}{v} \). The rule is pivotal when dealing with rational functions, like \( f(x) = \frac{x}{x+1} \), in our exercise.

The quotient rule states that if you have a function \( f(x) = \frac{u(x)}{v(x)} \), its derivative \( f'(x) \) can be found using:\[f'(x) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}\]

To simplify things:

• Identify where \( u(x) = x \) and \( v(x) = x + 1 \).
• Differentiating these gives \( u'(x) = 1 \) and \( v'(x) = 1 \).
Plug these into the quotient rule formula for simplicity. This approach breaks down complicated expressions into manageable parts, ensuring the efficient calculation of derivatives for rational functions.

In our specific case, this results in \( f'(x) = \frac{1}{(x+1)^2} \), giving us the foundational derivative needed to explore further analyses.

Linear approximation is a method used to estimate the value of a function using its tangent line at a particular point. It provides a simple way to approximate values of more complex functions locally, particularly near the point of interest. This method can simplify calculations, making them more tractable.

The formula for linearization is:\[L(x) = f(a) + f'(a)(x - a)\]where:

• \( f(a) \) is the function value at \( a \).
• \( f'(a) \) is the derivative at \( a \).
• \( x \) is the point at which we want to approximate \( f(x) \).

In the exercise, we used this at \( a = 1 \) to linearly approximate \( f(x) = \frac{x}{x+1} \) around \( x = 1.3 \). By substituting \( f(1) = \frac{1}{2} \) and \( f'(1) = \frac{1}{4} \) into our linear approximation formula, we simplified it to \( L(x) = \frac{1}{4}x + \frac{1}{4} \). This formula then helps us evaluate the behavior of \( f(x) \) near \( a \) without the more cumbersome calculations involved in determining \( f(x) \) itself.