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The Chain Rule is a fundamental tool in calculus used to compute the derivative of a composite function. When dealing with functions such as \(w = (uv - 1)^3\), the chain rule allows us to differentiate the function with respect to one of its variables while considering the inner expressions. First, we identify \(z = uv - 1\) as our inner function, which turns our original function into an easier form, \(w = z^3\). This simplification enables us to apply standard differentiation rules comfortably.

• Differentiate the outer function: \(\frac{\partial w}{\partial z} = 3z^2\)
• Differentiate the inner function with respect to the chosen variable: \(\frac{\partial z}{\partial u} = v\) or \(\frac{\partial z}{\partial v} = u\)

The Chain Rule then combines these into: \(\frac{\partial w}{\partial u} = \frac{\partial w}{\partial z} \cdot \frac{\partial z}{\partial u} = 3z^2 \cdot v\). Similarly, \(\frac{\partial w}{\partial v} = 3z^2 \cdot u\). Finally, substitute back \(z = uv - 1\) for complete expression.

Multivariable Calculus extends traditional calculus into functions of more than one variable. In the context of our exercise, understanding partial derivatives is crucial as it involves distinguishing how a function changes with respect to each variable independently. For the function \(w = (uv - 1)^3\), you focus on each variable one at a time.

• Partial derivative with respect to \(u\): \(\frac{\partial w}{\partial u} = 3v(uv - 1)^2\)
• Partial derivative with respect to \(v\): \(\frac{\partial w}{\partial v} = 3u(uv - 1)^2\)

Each partial derivative treats all other variables as constants. Multivariable Calculus empowers us to explore changes across multiple dimensions, each of which can influence the function's behavior in unique ways.

Differentiating a function involves determining its rate of change. In our exercise, we find how \(w\) changes with respect to \(u\) and \(v\). Differentiation is a fundamental concept in calculus that helps us understand the underlying behavior of functions. To effectively handle function differentiation in this context:- Identify dependent and independent variables.- Use the Chain Rule when functions have nested parts, as demonstrated when we set \(z = uv - 1\).Here, differentiating \(w = (uv - 1)^3\) with respect to \(u\) involves treating \(v\) as a constant:\[\frac{\partial w}{\partial u} = 3v(uv - 1)^2\]Similarly, differentiating with respect to \(v\) while treating \(u\) as constant yields:\[\frac{\partial w}{\partial v} = 3u(uv - 1)^2\]These computations showcase how differentiation with respect to each variable isolates the specific influences each has on the function \(w\).