Question

Find all the second partial derivatives in each of the following cases: \(\begin{aligned}&\text { (a) } z=\ln x(\mathrm{~b}) z=\ln y & (\mathrm{c}) z=\ln x y\end{aligned}\) (d) \(z=x \ln y\) (e) \(z=y \ln x\)

Short answer

Answer: (a) \(\frac{\partial^2 z}{\partial x^2} = -\frac{1}{x^2}\), \(\frac{\partial^2 z}{\partial x \partial y} = 0\), \(\frac{\partial^2 z}{\partial y^2} = 0\), \(\frac{\partial^2 z}{\partial y \partial x} = 0\). (b) \(\frac{\partial^2 z}{\partial x^2} = 0\), \(\frac{\partial^2 z}{\partial x \partial y} = 0\), \(\frac{\partial^2 z}{\partial y^2} = -\frac{1}{y^2}\), \(\frac{\partial^2 z}{\partial y \partial x} = 0\). (c) \(\frac{\partial^2 z}{\partial x^2} = -\frac{1}{x^2}\), \(\frac{\partial^2 z}{\partial x \partial y} = 0\), \(\frac{\partial^2 z}{\partial y^2} = -\frac{1}{y^2}\), \(\frac{\partial^2 z}{\partial y \partial x} = 0\). (d) \(\frac{\partial^2 z}{\partial x^2} = 0\), \(\frac{\partial^2 z}{\partial x \partial y} = \frac{1}{y}\), \(\frac{\partial^2 z}{\partial y^2} = -\frac{x}{y^2}\), \(\frac{\partial^2 z}{\partial y \partial x} = \frac{1}{y}\). (e) \(\frac{\partial^2 z}{\partial x^2} = -\frac{y}{x^2}\), \(\frac{\partial^2 z}{\partial x \partial y} = \frac{1}{x}\), \(\frac{\partial^2 z}{\partial y^2} = 0\), \(\frac{\partial^2 z}{\partial y \partial x} = \frac{1}{x}\).

Step-by-step solution

(a) Case z = ln(x)

For z = ln(x), First, we need to take the first partial derivatives with respect to 'x' and 'y': \(\frac{\partial z}{\partial x} = \frac{1}{x}\), \(\frac{\partial z}{\partial y} = 0\). Now, let's take the second partial derivatives of these derivatives with respect to 'x' and 'y': \(\frac{\partial^2 z}{\partial x^2} = -\frac{1}{x^2}\), \(\frac{\partial^2 z}{\partial x \partial y} = 0\), \(\frac{\partial^2 z}{\partial y^2} = 0\), \(\frac{\partial^2 z}{\partial y \partial x} = 0\).

(b) Case z = ln(y)

For z = ln(y), First, we need to take the first partial derivatives with respect to 'x' and 'y': \(\frac{\partial z}{\partial x} = 0\), \(\frac{\partial z}{\partial y} = \frac{1}{y}\). Now, let's take the second partial derivatives of these derivatives with respect to 'x' and 'y': \(\frac{\partial^2 z}{\partial x^2} = 0\), \(\frac{\partial^2 z}{\partial x \partial y} = 0\), \(\frac{\partial^2 z}{\partial y^2} = -\frac{1}{y^2}\), \(\frac{\partial^2 z}{\partial y \partial x} = 0\).

(c) Case z = ln(xy)

For z = ln(xy), First, we need to take the first partial derivatives with respect to 'x' and 'y': \(\frac{\partial z}{\partial x} = \frac{y}{xy} = \frac{1}{x}\), \(\frac{\partial z}{\partial y} = \frac{x}{xy} = \frac{1}{y}\). Now, let's take the second partial derivatives of these derivatives with respect to 'x' and 'y': \(\frac{\partial^2 z}{\partial x^2} = -\frac{1}{x^2}\), \(\frac{\partial^2 z}{\partial x \partial y} = 0\), \(\frac{\partial^2 z}{\partial y^2} = -\frac{1}{y^2}\), \(\frac{\partial^2 z}{\partial y \partial x} = 0\).

(d) Case z = xln(y)

For z = xln(y), First, we need to take the first partial derivatives with respect to 'x' and 'y': \(\frac{\partial z}{\partial x} = \ln(y)\), \(\frac{\partial z}{\partial y} = \frac{x}{y}\). Now, let's take the second partial derivatives of these derivatives with respect to 'x' and 'y': \(\frac{\partial^2 z}{\partial x^2} = 0\), \(\frac{\partial^2 z}{\partial x \partial y} = \frac{1}{y}\), \(\frac{\partial^2 z}{\partial y^2} = -\frac{x}{y^2}\), \(\frac{\partial^2 z}{\partial y \partial x} = \frac{1}{y}\).

(e) Case z = yln(x)

For z = yln(x), First, we need to take the first partial derivatives with respect to 'x' and 'y': \(\frac{\partial z}{\partial x} = \frac{y}{x}\), \(\frac{\partial z}{\partial y} = \ln(x)\). Now, let's take the second partial derivatives of these derivatives with respect to 'x' and 'y': \(\frac{\partial^2 z}{\partial x^2} = -\frac{y}{x^2}\), \(\frac{\partial^2 z}{\partial x \partial y} = \frac{1}{x}\), \(\frac{\partial^2 z}{\partial y^2} = 0\), \(\frac{\partial^2 z}{\partial y \partial x} = \frac{1}{x}\).

Key concepts

Partial Differentiation

Partial differentiation is like taking a specific slice of a bigger picture. Imagine you're looking at a hill, not as a whole, but from one particular direction at a time. With partial derivatives, we focus on how a function changes in relation to one variable, keeping the other variables constant.

When we say \(\frac{\partial z}{\partial x}\), it means we're adjusting only \(x\) while ignoring changes in \(y\). This helps us understand how \(z\) behaves if \(x\) changes just a bit. Similarly, \(\frac{\partial^2 z}{\partial x^2}\) means taking this process a step further, analyzing how the rate of change itself changes with respect to \(x\).

Second partial derivatives can be intriguing. You often end up finding mixed partial derivatives, such as \(\frac{\partial^2 z}{\partial x \partial y}\), indicating how the change in \(z\) concerning \(x\) is affected as \(y\) changes. They unlock more layers of understanding a function's behavior in multiple dimensions.

Functions of Multiple Variables

Many real-world situations depend on more than one factor. Think of a recipe—each ingredient affects the taste, but so does the way they interact. Functions of multiple variables are like that. They're mathematical expressions that result in a single output but depend on several inputs.

For example, with a function \(z = f(x, y)\), both \(x\) and \(y\) influence \(z\). Each variable provides a dimension to think about. If you imagined a three-dimensional graph, \(x\) and \(y\) could be your two horizontal directions, while \(z\) represents the height at each point. This gives you a surface, called a surface plot. Such plots help us visualize how different variables interact, showing peaks and valleys where values change.

With partial derivatives, we analyze these functions more deeply. They help us understand spots on the surface where different directions of change occur. Whether you're climbing a hill, understanding temperature variations, or assessing financial risks, controlling one variable at a time gives valuable insight into complex systems.

Engineering Mathematics

Engineering mathematics involves a rich interplay of mathematics with engineering problems. Mathematicians often use advanced calculus, such as partial differentiation, to break down and solve complex problems.

In engineering, one often deals with systems influenced by many factors simultaneously. Consider fluid dynamics, where the speed and direction of fluid flow may be influenced by pressure and temperature in various directions. By employing functions of multiple variables and partial derivatives, engineers can predict how changes in one factor affect the whole system.

For instance, in the design of a car's engine, there are myriad variables to consider, such as fuel type, cylinder movement, and air intake. F.E.A. (Finite Element Analysis) uses these principles to accurately model physical behavior. Second partial derivatives become crucial in ensuring stability and optimization of structures and processes. Hence, understanding how changes in one part of a system might influence the whole aids engineers in crafting safer, more efficient designs.