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In multivariable calculus, the concept of partial derivatives is crucial. It involves finding the derivative of a function with respect to one variable while keeping other variables constant. This is unlike single-variable calculus, where changes are considered with respect to only one independent variable.
For a function like \(z = xe^{x+y}\), partial derivatives allow us to understand how the function changes as each independent variable (in this case, \(x\) and \(y\)) changes, individually. The notation \(\frac{\partial z}{\partial x}\) signifies the partial derivative of \(z\) with respect to \(x\).

Partial derivatives shed light on how \(z\) behaves:

• When you take the partial derivative of \(z\) with respect to \(x\), it tells you how \(z\) changes as \(x\) changes, holding \(y\) constant.
• Similarly, the partial derivative with respect to \(y\), \(\frac{\partial z}{\partial y}\), shows how \(z\) changes with \(y\), holding \(x\) constant.

The product rule is a fundamental technique in calculus used when differentiating functions that are products of two simpler functions. When dealing with partial derivatives, the product rule becomes essential, as seen in our example function: \(z = xe^{x+y}\).

To apply the product rule, consider \(z = u\times v\), where \(u = x\) and \(v = e^{x+y}\). The product rule states:

• The derivative of \(uv\) with respect to \(x\) is \(u\frac{\partial v}{\partial x} + v\frac{\partial u}{\partial x}\).

Using this principle,

• \(\frac{\partial z}{\partial x} = e^{x+y} + xe^{x+y}\).
• First, differentiate \(e^{x+y}\) with respect to \(x\), treating \(y\) as a constant, which yields \(e^{x+y}\).
• Then add the product of \(x\) and the same derivative, \(e^{x+y}\), showcasing the actual application of the rule.

This results in: \(\frac{\partial z}{\partial x} = e^{x+y}(1+x)\). This illustrates how the product rule helps in breaking down and differentiating complex functions.

Multivariable calculus is an extension of the basic calculus principles into functions with multiple variables. Unlike single-variable calculus, which deals with one independent variable, multivariable calculus considers scenarios where functions depend on two or more variables.

The function \(z = xe^{x+y}\) operates in a world of multivariable calculus, making concepts like partial derivatives and total differentials extremely relevant. Here's why multivariable calculus matters:

• It allows us to explore how functions change with individual variables in complex spaces.
• Understanding these changes require tools like partial derivatives, which we covered before.

Calculating the total differential \(dz\) in the exercise is a step further into multivariable calculus:

• It combines the individual effects of changes in each variable (\(x\) and \(y\) in this case) on the function \(z\).
• The formula for the total differential \(dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy\) encapsulates how changes in multiple directions combine to affect the function's overall change.

Multivariable calculus offers these unique insights, vital for more advanced applications in fields like physics, engineering, and economics.