91Ó°ÊÓ

In calculus, partial derivatives represent how a function changes as one of its independent variables is varied, while all other variables are held constant. When you have a function with several variables, like a multivariable function, each variable can affect the output independently.
In our exercise, the function we are dealing with is noted as \( z = f(x^2 + y^2) \) where \( u = x^2 + y^2 \). To find how this function changes with respect to each variable, we compute the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).
The chain rule assists in finding these partial derivatives. The expression becomes:

• \( \frac{\partial z}{\partial x} = \frac{df}{du} \cdot \frac{\partial u}{\partial x} \)
• \( \frac{\partial z}{\partial y} = \frac{df}{du} \cdot \frac{\partial u}{\partial y} \)

This method allows us to see how the function changes with respect to each of the variables, providing insight into its geometric and dynamic properties.

A differentiable function is one that has a derivative at every point in its domain. Simply put, this means you can find a tangent line at any point on the curve of the function, which represents the function's slope at that point. Being differentiable implies continuity, but it's a bit more specific.
In our exercise, it is given that \( f \) is a differentiable function of one variable. This characteristic ensures smoothness in the curve of \( f \), allowing us to compute derivatives without encountering points of discontinuity or sharp turns. When considering \( z = f(x^2 + y^2) \), this property extends to our composite function, meaning we can work with its derivatives reliably.

• Differentiability means the ability to apply derivative techniques smoothly.
• This property is crucial for computations involving composition of functions.

By confirming \( f \) and its composed form of \( z \) are differentiable, we can confidently apply the chain rule to determine partial derivatives.

Composite functions are functions within functions. They can be more complex, but the idea is straightforward: you have one function inside another. In many problems, including our exercise, dealing with composite functions is common because it allows a single dependent variable to depend on multiple independent ones simultaneously.
In this scenario, \( z = f(u) \) and \( u = x^2 + y^2 \). The composition ensures the behavior of \( z \) is influenced by both \( x \) and \( y \). A composite function is crucial when you want to express affects brought on by multiple variables succinctly.

• Lets \( z \) depend on more than direct input variables — expanding applicability.
• Simplifies influence of complex relationships between variables when viewing output.

Using composite functions, we can trace outcomes of variable changes through a more intricate chain of relations, affording us a broader view of function behavior in multidimensional contexts.