Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the second partial derivatives \(f_{x x^{\prime}} f_{x y^{\prime}} f_{y y^{\prime}}\) and \(f_{y x}\) at the point. $$ f(x, y)=\ln (x-y) $$

Step-by-Step Solution

Answer

The second partial derivatives of the function \(f(x, y)=\ln (x-y)\) are \(f_{xx} = 0\), \(f_{xy} = f_{yx} = \frac{1}{{(x - y)}^2}\), and \(f_{yy} = 0\). To evaluate these at a given point \((a, b)\), replace \(x\) and \(y\) with \(a\) and \(b\) respectively.

Step 1 - Calcuate the first partial derivatives

Using the power rule for differentiation, the first partial derivative with respect to \(x\) is \(f_{x} = \frac{1}{x - y}\), and the first partial with respect to \(y\) is \(f_{y} = -\frac{1}{x - y}\).

Step 2 - Calcuate the second partial derivatives

The second partial derivatives can be obtained by differentiating the first partial derivatives. Hence: \(f_{xx} = \frac{\partial}{\partial x}(f_{x}) = 0\), \(f_{xy} = \frac{\partial}{\partial y}(f_{x}) = \frac{1}{{(x - y)}^2}\), \(f_{yx} = \frac{\partial}{\partial x}(f_{y}) = \frac{1}{{(x - y)}^2}\), \(f_{yy} = \frac{\partial}{\partial y}(f_{y}) = 0\).

Step 3 - Evaluate the second partial derivatives at the given point

The exercise does not provide a specific point to evaluate these values at. However, should a point be provided, call it \((a, b)\), replace \(x\) with \(a\) and \(y\) with \(b\) in the second partial derivatives.

Key Concepts

Partial Differentiation

Partial differentiation is a fundamental technique used in multivariable calculus to determine how a function changes as each variable changes, while holding the other variables constant. In context with our exercise, we are examining the function \(f(x, y) = \ln(x - y)\), which depends on two variables, \(x\) and \(y\). When we find \(f_x\), the first partial derivative of the function with respect to \(x\), we treat \(y\) as a constant, and vice versa for \(f_y\). This allows us to analyze how the function behaves as we vary one independent variable at a time.

The process of finding a partial derivative is similar to finding a standard derivative in single-variable calculus. One key difference, though, is that we often deal with multiple first partial derivatives, as every independent variable gives rise to its own partial derivative. These are essential in constructing the tangent plane at a point on a surface, and they also lead to higher-order partial derivatives, which tell us about the concavity and interactions between variables.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of multiple variables. This field is pivotal in analyzing mathematical models with several degrees of freedom, which are prevalent in physics, engineering, and economics. Like single-variable calculus, multivariable calculus involves differentiation and integration, but it also introduces new concepts like gradient, divergence, and curl.

In our exercise focusing on second partial derivatives, we are delving into one of the main operations in multivariable calculus. By finding the second partial derivatives \(f_{xx}\), \(f_{xy}\), \(f_{yx}\), and \(f_{yy}\), we can infer crucial information about the curvature of the function's graph and predict the function's behavior around a given point. These derivatives are also useful for checking the function's concavity and for optimizing functions with respect to two or more variables.

First Partial Derivatives

First partial derivatives represent the rate at which a function changes as one of its variables changes, with all other variables held constant. In the case of our exercise, the first partial derivatives \(f_{x}\) and \(f_{y}\) are computed to understand how the function \(f\) responds to small changes in \(x\) and \(y\) independently. \(f_{x} = \frac{1}{x - y}\) indicates the rate of change of \(f\) in respect to \(x\), and \(f_{y} = -\frac{1}{x - y}\) describes the rate of change in respect to \(y\).

The calculation of these derivatives is a necessary step before finding the second partial derivatives. The ability to accurately compute first partial derivatives is crucial as it sets the groundwork for deeper analysis involving higher-order derivatives and is instrumental in applications such as gradient descent for optimization and in constructing differential equations governing physical phenomena.

Logarithmic Differentiation

Logarithmic differentiation is a technique that simplifies the derivative-taking process, especially for functions where direct differentiation is complex. It is particularly useful when dealing with functions involving products, quotients, powers, or functions of functions. Our exercise uses logarithmic differentiation implicitly because the function itself is a logarithm. The derivative of a natural logarithm \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\), and this concept carries over to the partial differentiation of \(f(x, y) = \ln(x - y)\).

Logarithmic differentiation provides a direct method to differentiate logarithmic functions with respect to one or more variables. It simplifies the process and reduces the chance of making algebraic errors. In a broader context, the technique can also be employed to differentiate functions where taking the logarithm at the start makes the differentiation process much more manageable.