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The chain rule is a fundamental tool in calculus that enables you to compute the derivative of a composite function. This means it helps find how fast a function is changing, even when it has functions inside it or made up of other functions.
When you have a function like \( f(x) = (u)^n \) where \( u \) itself is another function (for example, \( u = \frac{x^2 + 1}{x + 2} \) in our case), the chain rule allows you to break it down into easier parts for differentiation.
The rule is defined as: if \( y = (u)^n \), then the derivative \( \frac{dy}{dx} = n(u)^{n-1} \cdot \frac{du}{dx} \).

• We first differentiate the outer function with respect to \( u \), treating \( u \) as a single variable. For example, \( 3u^2 \) when \( y = u^3 \).
• Then we multiply the result by the derivative of the inner function \( u \) with respect to \( x \), or \( \frac{du}{dx} \).

This cascading process makes it easier to handle even the most complex functions by breaking them down into manageable components.

The quotient rule is a technique for finding the derivative of a fraction of two functions. In simpler terms, it helps us differentiate expressions where one function divides another.
For a function \( u(x) = \frac{v(x)}{w(x)} \), the rule is mathematically expressed as: \[ \frac{du}{dx} = \frac{v'(x)w(x) - v(x)w'(x)}{[w(x)]^2} \]
Here's how it works practically:

• Identify the top and bottom parts of the fraction. Here, \( v = x^2 + 1 \) and \( w = x + 2 \).
• Calculate their derivatives, \( v'(x) = 2x \) and \( w'(x) = 1 \).
• Plug into the formula to find \( \frac{du}{dx} \), and simplify.

The quotient rule deftly handles the division of functions, providing an elegant solution even when both numerator and denominator are complex expressions.

Once we find the derivative of a function, the next step is often to evaluate it at a specific point. This involves plugging a number into the derivative to find the rate of change at that specific value.
In the given exercise, we determine \( f'(x) \) and then evaluate it at \( x = 3 \). Steps include:

• Substitute the number into each part of the derivative expression. For instance, where \( x = 3 \) in \( f'(x) = 3 \left(\frac{x^2+1}{x+2}\right)^2 \cdot \frac{x^2 + 4x - 1}{(x+2)^2} \).
• Simplify each component separately. Start with the inside functions and work outwards to streamline calculations.
• Multiply through to find the final value of \( f'(3) \).

Derivative evaluation is particularly useful in practical scenarios, such as calculating the instantaneous speed of a moving object at a particular time, ensuring understanding goes beyond just theoretical exercise.