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The product rule is a fundamental technique in calculus used to differentiate expressions where two functions are multiplied together.
This rule is particularly essential when both functions are variables that change over time, such as the length and width of a rectangle in our example.
The formal statement of the product rule is as follows: If you have two functions, say, \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

• \( \frac{d}{dx}[u(x) \, \cdot \, v(x)] = u(x) \cdot \frac{dv}{dx} + v(x) \cdot \frac{du}{dx} \)

This means you take the derivative of the first function and multiply by the second, then add it to the product of the first function and the derivative of the second.
In our context, where the area \( A = L \times W \), we use the product rule because both length \( L \) and width \( W \) change over time.
Applying the product rule helps us correctly find how fast the area is changing by considering how each dimension contributes to the change.

Differentiation is a process in calculus that involves finding the rate at which a quantity changes.
This is crucial for solving problems involving dynamic systems where multiple variables are changing, like in our rectangle.
Differentiation gives us a tool to compute things like velocity, by understanding how a certain measurement changes over time.
When dealing with problems like this one, it's about figuring out the relationships between various rates.
To differentiate the area of the rectangle \( A \) concerning time \( t \), we apply the product rule as part of differentiation:

• We differentiate \( A = L \times W \) with respect to time, \( t \), giving us \( \frac{dA}{dt} = L \frac{dW}{dt} + W \frac{dL}{dt} \).

This expression represents how the rectangle's area changes over time, by considering both the rate of change of the length and the width.

Rectangles are simple geometric shapes that have opposite sides equal and parallel.
The properties of rectangles are fundamental to solving problems that involve area and perimeter calculations.
In our context, the formula for the area of a rectangle, \( A = L \times W \), is especially useful.
This exercise involves understanding how changes in the dimensions of a rectangle can affect its overall area.
The length \( L \) and width \( W \) of the rectangle are both dynamic and changing over time. So, calculating how these changes impact the area becomes an application of calculus techniques like differentiation and the product rule.
The insight we get from these calculations can help us predict things like the growth of an area given certain changes.

The concept of the rate of change is synonymous with how fast something is changing over time.
It's a core idea in calculus, generally related to derivatives, that helps us understand the dynamics of changing systems.
In this exercise, we are given the rates at which the length and width of a rectangle change:

• The length increases at \( 8 \) cm/s, and the width increases at \( 3 \) cm/s.

These values help predict how the area changes, which is what we refer to as the rate of change of the area.
Through differentiation, we determine this rate:

• Inserting known values into the form \( \frac{dA}{dt} = 20 \times 3 + 10 \times 8 \), gives the final rate of \( 140 \) cm²/s.

Ultimately, understanding the rate of change gives us invaluable information about how one quantity affects others over time, valuable in forecasting and decision-making.