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The Product Rule is essential when dealing with derivatives of functions that are products of two functions. It's like a tool for untangling knots made by two interwoven functions. If you have two functions, say \( u(x) \) and \( v(x) \), and you need to differentiate their product, the product rule provides the solution. It states that the derivative of the product \( uv \) is given by the formula:

• \( (uv)' = u'v + uv' \)

In simpler terms, you take the derivative of one function while keeping the other constant and vice versa. Then, just add those two results together.
To illustrate, consider the function \( f(x) = x e^{x} \). In this case, set \( u(x) = x \) and \( v(x) = e^x \). Their derivatives are straightforward: \( u'(x) = 1 \) and \( v'(x) = e^{x} \).
By applying the product rule, we get:

• \( f'(x) = 1 \,\cdot\, e^{x} + x \,\cdot\, e^{x} \)
• Which simplifies to \( f'(x) = e^{x} (1 + x) \)

Learning the product rule is like having a reliable method to handle complex derivatives, a key skill in calculus.

Critical points are crucial for understanding where a function's behavior changes. A critical point occurs where the derivative of the function equals zero or is undefined. These are the spots where the function might have a maximum, a minimum, or a saddle point.
For our function \( f(x) = x e^x \), the derivative \( f'(x) = e^{x}(1 + x) \). We're interested in finding where this derivative equals zero since \( e^{x} \) is never zero.
Therefore, we solve:

• \( 1 + x = 0 \)
• This gives \( x = -1 \)

Here, \( x = -1 \) is our critical point, a potential location for a local extremum within the interval \([-\frac{3}{2}, 1] \).
Identifying critical points is the first step in analyzing the behavior of functions, especially for making observations about local extrema.

Local extrema refer to the peaks and valleys in the graph of a function. A local extremum is a local maximum or minimum. These are important because they show where the function changes direction in a small section of the graph.
In our example of \( f(x) = x e^{x} \), the critical point \( x = -1 \) presents a change in direction.
To determine the type of extremum, we can observe the signs of the derivative around this point. This is known as the first derivative test.

• If \( f'(x) \) changes from negative to positive as \( x \) moves through the critical point, then the function has a local minimum at that point.
• Conversely, if \( f'(x) \) changes from positive to negative, it has a local maximum there.

In the derivative \( f'(x) = e^{x} (1 + x) \):

• For \( x < -1 \), \( f'(x) < 0 \) indicating decreasing function.
• For \( x > -1 \), \( f'(x) > 0 \) showing an increasing function.

Thus, at \( x = -1 \), the function has a local minimum.
This local behavior gives insights into the function's form without needing to analyze every possible value.

Sign analysis is a helpful method for understanding how a function behaves across its domain. It involves checking the sign (positive or negative) of a derivative in given intervals to predict the function's trends of increase or decrease.
To conduct sign analysis for \( f(x) = x e^x \), we use its derivative: \( f'(x) = e^{x} (1 + x) \).
We'll examine the sign changes across the intervals around the critical point:

• For \( x < -1 \), \( 1 + x < 0 \), thus \( f'(x) < 0 \). This tells us the function decreases.
• At \( x = -1 \), \( f'(x) = 0 \), a critical point where the function stops decreasing and might change direction.
• For \( x > -1 \), \( 1 + x > 0 \), so \( f'(x) > 0 \). Here, \( f(x) \) increases.

Performing a sign analysis offers a visual understanding of how the function behaves before, at, and after a critical point. This gives a more extensive map of the behavior pattern of functions on an interval and highlights where extrema might lie.