91Ó°ÊÓ

The Chain Rule is a fundamental concept in calculus that helps us differentiate composite functions. More simply, it's about finding the rate of change of one variable concerning another.
When dealing with related rates, like in our problem, we can use the chain rule to connect the rates of change between different variables.
Here's how it works:

• Identify the original function, such as the given function \( y = x^2 \).
• Apply the chain rule formula: \( \frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt} \).

This formula helps us understand the relationship between the rate of change of \( y \) with respect to time \( t \) (\( \frac{dy}{dt} \)), and the rate of change of \( x \) with respect to time \( t \) (\( \frac{dx}{dt} \)), leveraging the immediate transformation between \( y \) and \( x \) (\( \frac{dy}{dx} \)).
By using the chain rule, we effectively break down a complicated process into easier steps where each piece logically follows the next.

A derivative represents the rate of change of a variable. It is the foundation of differential calculus.
In our example, we need to find the derivative of \( y \) with respect to \( x \), which is the instantaneous rate of change of \( y \) for a small increment in \( x \).

• The given function is \( y = x^2 \).
• The derivative of this function, \( \frac{dy}{dx} \), measures how \( y \) changes as \( x \) changes, calculated by taking \( 2x \).

Derivatives help us understand how one quantity affects another. They are vital when calculating speed, growth rates, and other dynamic processes.
In our case, knowing the derivative allows us to find out how fast \( y \) is changing in relation to time \( t \). This is crucial for solving problems involving dynamic systems or moving objects.

Differentiating a function involves a series of steps that transform a given function into its derivative, allowing us to study how it changes over time.
Let's break down the differentiation steps for this problem:

• **Given Function**: Start with the function \( y = x^2 \).
• **Compute \( \frac{dy}{dx} \)**: Differentiate \( y \) with respect to \( x \). Here, the derivative of \( x^2 \) is \( 2x \).
• **Apply Chain Rule**: Use the formula \( \frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt} \). This allows us to link \( \frac{dy}{dt} \) with \( \frac{dx}{dt} \), given that \( \frac{dx}{dt} = 3 \).
• **Substitute and Solve**: Plug in \( x = -1 \), yielding \( \frac{dy}{dt} = 2(-1) \times 3 \).
• **Result**: Solve to find \( \frac{dy}{dt} = -6 \).

The process is structured to simplify complex functions, using both derivative calculations and substitutions to reach useful conclusions.
Following these steps ensures that you methodically address all aspects of a differentiation problem, giving you accurate and reliable results.