91Ó°ÊÓ

Partial derivatives are essential tools in multivariable calculus. They allow us to understand how a function changes as one of its variables changes, while all other variables are held constant. For a function of multiple variables, like \( z = e^{xy} \tan y \), a partial derivative with respect to \( x \) would measure the rate of change of \( z \) when there is a small change in \( x \), keeping \( y \) constant.

This concept is crucial because it gives insight into the slope of the function in the direction of the variable of interest. In our calculation:

• \( \frac{\partial z}{\partial x} = e^{(xy)} \tan{y} \cdot y \) tells us how \( z \) changes with \( x \).
• \( \frac{\partial z}{\partial y} = e^{(xy)} \sec^2{y} \cdot x + e^{(xy)} \tan{y} \) indicates the change in \( z \) with \( y \).

Exploring these partial derivatives helps us understand the behavior of the function in a multidimensional space, which is the core of multivariable calculus.

Multivariable calculus extends the concepts of single-variable calculus to functions with more than one variable. It deals with deriving and analyzing formulas that involve multiple variables. Common processes include differentiation, integration, and optimization in higher dimensions.

One of the major challenges in multivariable calculus is understanding how multiple variables collectively influence changes in a function. For instance, in our study with \( x = 4s + 5t \) and \( y = \frac{3s}{3t} \), each variable \( s \) and \( t \) affects both \( x \) and \( y \), making it complex to see how they change \( z \) simultaneously.

By using techniques like partial derivatives, we can study each path of influence independently. Additionally, calculus tools like the chain rule bridge these relationships by explaining how changes in the input variables \( s \) and \( t \) affect the output \( z \). This intricate understanding aids in solving real-world problems involving multiple variables, spanning fields such as physics, engineering, and economics.

Solving calculus problems, especially involving multivariable functions, requires careful application of computational rules and a clear understanding of the underlying concepts. The step-by-step solution involves:

- **Identifying Functions and Relationships**: Recognize the primary function, \( z = e^{xy} \tan y \), and rewrite it to see which variables influence one another, expressed here as \( x = 4s + 5t \) and \( y = \frac{3s}{3t} \).
- **Differentiating Using Partial Derivatives**: Determine how \( z \) changes with each of the embedded variables \( x \) and \( y \) by computing \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).
- **Application of the Chain Rule**: Implement the chain rule for derivatives to interconnect the changes from \( s \) and \( t \) to \( z \) via their relationships through \( x \) and \( y \). This involves finding \( \frac{\partial x}{\partial s}, \frac{\partial x}{\partial t}, \frac{\partial y}{\partial s}, \) and \( \frac{\partial y}{\partial t} \), and then combining them through the formula:

• \( \frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s} \)
• \( \frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t} \)

This structured approach, breaking complex problems into manageable pieces, allows us to tackle challenging calculus problems successfully.