Question

Find the first partial derivatives with respect to \(x\) and with respect to \(y\). $$ g(x, y)=e^{x / y} $$

Short answer

The first partial derivatives of \(g(x, y)=e^{x / y}\) are \(\frac{\partial g(x, y)}{\partial x} = \frac{1}{y}e^{x/y}\) and \(\frac{\partial g(x, y)}{\partial y} = -\frac{x}{y^2}e^{x/y}\)

Step-by-step solution

Differentiate with respect to \(x\)

This is a function of a function, or composite function. Specifically, it is of the form \(f(g(x))\), where \(f(u) = e^u\) and \(g(x) = x/y\). Finding the derivative of this using the Chain Rule gives:\[\frac{\partial g(x, y)}{\partial x} = e^{x/y} \cdot \frac{1}{y}\]As the derivative of \(x/y\) with respect to \(x\) is \(1/y\). The first part of \(e^{x/y}\) remains the same as the derivative of \(e^x\) is itself.

Differentiate with respect to \(y\)

We use the Chain Rule again. However, the derivative of \(x/y\) with respect to \(y\) is \(-x/y^2\). Therefore, you get:\[\frac{\partial g(x, y)}{\partial y} = e^{x/y} \cdot \(-\frac{x}{y^2}\)\]

Simplify Expressions

After performing the derivatives, we can simplify the results:\[\frac{\partial g(x, y)}{\partial x} = \frac{1}{y}e^{x/y}\]and \[\frac{\partial g(x, y)}{\partial y} = -\frac{x}{y^2}e^{x/y}\]

Key concepts

Chain Rule

The chain rule in calculus is a formula used to calculate the derivative of a composite function. In simpler terms, it helps us find the rate of change in situations where one variable affects another through a function within a function.

For example, consider a function that describes temperature over time as it increases in a pattern that itself changes over time. If we wanted to know how quickly the temperature rises in relation to time, we'd be dealing with a composite function, and the chain rule would be the tool to use.

In the context of partial derivatives, the chain rule allows us to differentiate a function with respect to one variable, even when that function is defined in terms of another variable. This approach is essential when the function being differentiated is affected indirectly by the variable being considered.

Multivariable Calculus

Multivariable calculus extends the concept of calculus to functions of several variables. Unlike single-variable calculus where the focus is on understanding the behavior of functions as they relate to one dimension, multivariable calculus deals with functions that have inputs in two or more dimensions.

The concept explored in the exercise is a fundamental part of multivariable calculus called partial differentiation. Here we focus on the change in a function in relation to one variable at a time, while treating other variables as constants.

This is particularly useful when dealing with physical phenomena where different factors can influence an outcome, such as temperature variations across different points on a 2D surface. Understanding how a change in one dimension impacts the function without immediately considering other variables helps us isolate and study the effects systematically.

Exponential Functions Differentiation

Differentiation of exponential functions follows specific rules. Exponential functions are of the form \(f(x) = a^x\), where \(a\) is a constant. The derivative of an exponential function can be found using the base of the natural logarithm, \(e\). The unique property of the exponential function \(e^x\) is that the rate of change is equal to the value of the function itself at any point.

Thus, when differentiating \(e^x\), the derivative is \(e^x\). This rule simplifies calculations in cases where the exponential function is involved, and it's particularly important when the exponent includes a composite function, calling for the application of the chain rule.

Partial Differentiation

Partial differentiation is the process of deriving the rate of change of a multivariable function concerning one variable, while keeping other variables constant. This concept is essential in fields like engineering, physics, and economics, where multi-factor systems are common.

Partial derivatives are represented as \(\frac{\partial z}{\partial x}\) where \(z\) is a function of both \(x\) and \(y\), and we're interested in how \(z\) changes as \(x\) changes whilst \(y\) remains constant. The exercise demonstrates this concept thoroughly. By understanding how to apply partial differentiation, we can analyze the impact of single factors within a system, making it easier to predict and optimize outcomes by studying these individual effects.