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The chain rule is an essential tool in calculus, especially for dealing with complex functions that are composed of other functions. It helps us differentiate composite functions by breaking them down into their individual components. For a function like \( z = e^{k(r-x)} \), where \( r^{2}=x^{2}+y^{2} \), the chain rule allows us to differentiate each part with respect to a variable, such as \( x \) or \( y \).

Here's how it works:

• Start with the outermost function, which in this case is the exponential function \( e^u \) where \( u = k(r-x) \).
• Differentiating \( e^u \) with respect to \( u \) gives you the derivative \( e^u \), based on standard derivatives of exponential functions.
• Next, differentiate the innermost function \( u = k(r-x) \), with respect to the variable of interest, such as \( x \) or \( y \).

This step-by-step differentiation gives you the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \). Whenever you see functions within other functions, remember the chain rule can help you untangle them systematically.

The second derivative, noted by \( \frac{\partial^2 z}{\partial x^2} \) or \( \frac{\partial^2 z}{\partial y^2} \), represents the rate at which the first derivative of a function changes. For our function \( z = e^{k(r-x)} \), finding the second derivative can tell us how the behavior of \( z \) changes as \( x \) or \( y \) change.

When calculating the second derivative:

• You first find the first derivative, using methods such as the chain rule.
• Differentiate this first derivative a second time.

In this exercise, we find:

• \( \frac{\partial^2 z}{\partial x^2} = k^2z \left(1 + 2\frac{x}{r} + \frac{x^2}{r^2}\right) \)
• \( \frac{\partial^2 z}{\partial y^2} = k^2z \frac{y^2}{r^2} \)

This shows not only direct relationships but also how these expressions combine and simplify in conjunction with the partial differential equation given. Understanding second derivatives is crucial as they can reveal concavity, acceleration, and other dynamic aspects of the function.

Exponential functions, recognizable by the format \( e^{u} \), are integral parts of many mathematical models and equations. They have a unique property: the derivative of an exponential function \( e^u \) with respect to \( u \) is itself, \( e^u \). This makes differentiation straightforward once you've isolated the exponent.

For \( z = e^{k(r-x)} \), the exponential function directly influences how \( z \) changes with \( x \) and \( y \). They are important in this exercise because:

• They simplify multiplication and division due to their growth properties, which helps in solving differential equations.
• In combination with the chain rule, they help assess complex nested functions through a systematic approach to differentiation.

These reasons are why exponential functions are common in advanced calculus and physics, providing powerful tools for modeling growth and changes.

Mathematical proofs use logical reasoning to establish the truth of a mathematical statement. In this exercise, proofs are used to verify the given parts \( (a) \) and \( (b) \). Through substitution and simplification, we demonstrate that the identities hold true.

For part (a):

• Substitute the obtained expressions for \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).
• Simplify the equation through algebraic manipulation to establish it equals zero.

For part (b):

• Utilize second derivatives and previously solved expressions.
• Simplify calculations to match both sides of the equation given in the problem.

Proofs are a step-by-step affirmation of a mathematical truth, ensuring that every transition from one step to the next is based on valid mathematical principles. They form the backbone of mathematical theory and problem-solving.