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Multivariable calculus is a branch of mathematics that extends the principles of calculus to functions of multiple variables. Unlike single-variable calculus, which deals with functions governed by a single independent variable, multivariable calculus investigates functions dependent on several inputs. This expansion helps to explore complex systems that occur in areas like physics, engineering, and economics.
Multivariable calculus encompasses several key concepts:

• Partial Derivatives: Represents how a function changes as each variable is independently varied. For a function of two variables, like our example with \( z = x + f(u) \), partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \) describe the rate of change of \( z \) with respect to \( x \) and \( y \), respectively.
• Gradient: A vector containing all the partial derivatives of a function, which points in the direction of greatest increase of the function.
• Multiple Integrals: Allow integration over regions in space, generalizing the concept of a single integral over an interval.

In our exercise, calculating partial derivatives helps identify how changing one variable affects the output, proving crucial in solving the given problem.

The chain rule is a fundamental theorem in calculus used for finding the derivative of a composite function. In the context of multivariable calculus, the chain rule adapts to deal with functions where variables themselves are tied to other variables. This linkage allows for differentiation in more intricate settings, as seen in our problem where \( u = xy \) is a function of \( x \) and \( y \).
Using the chain rule involves several steps:

• Differentiate the outer function while treating the inner function as a constant. In our case, \( z = x + f(u) \).
• Multiply this result by the derivative of the inner function with respect to your variable of interest. This means finding \( \frac{df}{du} \cdot \frac{\partial u}{\partial x} \) when differentiating with respect to \( x \) and \( \frac{df}{du} \cdot \frac{\partial u}{\partial y} \) when differentiating with respect to \( y \).

In practice, this rule simplifies the process of finding derivatives for complex functions, enabling the solution of systems reliant on interconnected variables.

Differentiation is the process of finding the derivative of a function, which measures how a function's output changes as its inputs change. In the realm of multivariable calculus, differentiation extends to functions involving more than one input, requiring methods like partial differentiation and employing rules such as the chain rule.
Differentiation processes include:

• Ordinary Derivatives: Determines the rate of change of a function concerning a single variable.
• Partial Derivatives: Essential for dealing with functions with multiple inputs. They express the function's change concerning one variable at a time, keeping others constant.

Particularly in our exercise, differentiation demonstrates how alterations in \( x \) and \( y \) independently influence the value of \( z \). By calculating these individual effects using partial derivatives, we established the relationship outlined, meeting the problem's requirements. This analysis highlights differentiation's significance in understanding the sensitivity and behavior of multivariable functions.