91Ó°ÊÓ

The product rule is a fundamental concept in calculus, especially when dealing with derivatives. When you have a product of two functions and you need to differentiate them with respect to a variable, you use the product rule. The basic formula for the product rule is: \[ \frac{d}{dx}(u(x) \cdot v(x)) = u'(x)v(x) + u(x)v'(x) \]Here, \(u(x)\) and \(v(x)\) are two functions of \(x\). The derivative of the product is the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function. In our exercise, when finding the partial derivative of \(z = x e^{x / y}\) with respect to \(x\), we apply the product rule. The function \(x\) acts as \(u(x)\), and \(e^{x/y}\) is \(v(x)\), which gives us:- Derivative of \(x\) with respect to \(x\) is 1.- Derivative of \(e^{x/y}\) with respect to \(x\) is \(e^{x/y}/y\). Combine these using the product rule to find the first partial derivative. Remember, using this rule can simplify the process and avoid common errors when handling functions of products.

Exponential functions are a key concept often appearing in calculus and differential equations. They are denoted as \(e^x\) where \(e\) is a mathematical constant approximately equal to 2.71828. Their unique property is that the rate of growth of the exponential function is proportional to its value, making it fundamental in modeling growth processes.In the given function \(z = x e^{x/y}\), the term \(e^{x/y}\) is an exponential function. What makes it special here is that the exponent is a fraction, \(x/y\), involving two variables, \(x\) and \(y\). This setup is common in real-world problems where quantities vary with respect to two or more variables. When taking derivatives, it's important to remember that:- The derivative of \(e^{u}\) with respect to \(x\) is \(e^{u} \cdot u'\) where \(u\) is a function of \(x\).Understanding how exponential functions behave under differentiation is crucial for solving problems like the one in this exercise.

Functions of two variables, such as the one given in our exercise \(z = x e^{x/y}\), represent surfaces in three-dimensional space. Each pair of values from the variables \(x\) and \(y\) determines a unique value of \(z\). In this case, the function combines a linear term \(x\) and an exponential term \(e^{x/y}\).Partial derivatives allow us to understand how these surfaces change when we vary just one of the variables while keeping the other constant. This is why we calculate the partial derivative with respect to \(x\) and with respect to \(y\). Here's what happens when we partially differentiate:- With respect to \(x\), treat \(y\) as a constant.- With respect to \(y\), treat \(x\) as a constant.The outcomes, \(\frac{dz}{dx}\) and \(\frac{dz}{dy}\), describe how the value of \(z\) changes in response to small changes in \(x\) and \(y\) respectively. Visualizing this can help clarify the effect each variable has on the overall function. Functions of two variables are vital for modeling problems that depend on multiple inputs, and mastering partial derivatives is essential for deeper analysis.