Problem 48 - Calculus Early Transcendentals

Question

Assume that \(F(x, y, z(x, y))=0\) implicitly defines \(z\) as a differentiable function of \(x\) and \(y .\) Extend Theorem 9 to show that \(\frac{\partial z}{\partial x}=-\frac{F_{x}}{F_{z}}\) and \(\frac{\partial z}{\partial y}=-\frac{F_{y}}{F_{z}}\).

Step-by-Step Solution

Answer

Question: Determine the partial derivatives of the function \(z(x, y)\) defined implicitly by the equation \(F(x, y, z(x, y)) = 0\). Answer: The partial derivatives of the function z with respect to x and y are \(\frac{\partial z}{\partial x} = -\frac{F_{x}}{F_{z}}\) and \(\frac{\partial z}{\partial y} = -\frac{F_{y}}{F_{z}}\).

Step 1: Compute the total derivative of the given equation with respect to x and y

Start by finding the total derivative of the given equation \(F(x, y, z(x, y)) = 0\) with respect to x and y: For x: \(\frac{d}{dx}(F(x, y, z(x, y))) = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial z}\frac{\partial z}{\partial x} = 0\) For y: \(\frac{d}{dy}(F(x, y, z(x, y))) = \frac{\partial F}{\partial y} + \frac{\partial F}{\partial z}\frac{\partial z}{\partial y} = 0\)

Step 2: Solve for the partial derivatives of z with respect to x and y

Now, we will solve each equation for the partial derivative of z: For x: \(\frac{\partial z}{\partial x} = -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial z}} = -\frac{F_{x}}{F_{z}}\) For y: \(\frac{\partial z}{\partial y} = -\frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial z}} = -\frac{F_{y}}{F_{z}}\) Hence, the partial derivatives of the function z with respect to x and y are \(\frac{\partial z}{\partial x} = -\frac{F_{x}}{F_{z}}\) and \(\frac{\partial z}{\partial y} = -\frac{F_{y}}{F_{z}}\).

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in calculus, especially when dealing with functions of multiple variables. Imagine a function that depends on two variables: say, temperature at different points in a room depending on x (width) and y (height). A partial derivative, denoted as \( \frac{\partial}{\partial x} \) or \( \frac{\partial}{\partial y} \), measures how the function changes as one of the variables changes, while keeping other variables constant. This is a bit like observing how temperature changes if we move only sideways or upwards in an imaginary grid.

• Notation: They are often represented with the symbol \( \partial \), distinguishing them from ordinary derivatives.
• Meaning: Each partial derivative gives the rate of change of the function in one direction, parallel to one of the axes of input space.
• Example: For a function \( F(x, y, z) \), \( \frac{\partial F}{\partial x} \) refers to the rate of change of F as x changes, keeping y and z constant.

Understanding how each variable influences a multi-variable function is crucial, especially in scenarios involving complex systems or fields.

Differentiable Functions

A differentiable function is one that has a well-defined tangent at each point in its domain. Simply put, if a function is differentiable at a point, it means we can "zoom in" enough at that point to make it look like a straight line. In more advanced contexts, especially in multivariable calculus, differentiability implies that partial derivatives exist and are continuous.

• Smoothness: Differentiable functions tend to be smooth without any breaks, jumps, or sharp corners.
• Differentiation: The process of finding a derivative is called differentiation, and it plays a key role in understanding dynamics and changes in systems.
• Example: Consider the function \( z = x^2 + y^2 \). It’s differentiable across its entire domain, allowing us to calculate partial derivatives like \( \frac{\partial z}{\partial x} = 2x \) and \( \frac{\partial z}{\partial y} = 2y \).

Generalizing the idea from one dimension, differentiable functions in several variables are the backbone of topics such as gradient descent used in optimization problems.

Total Derivative

The total derivative is an extension of the concept of differentiation to functions of multiple variables. When the behavior of a function depends on several variables simultaneously, the total derivative provides a comprehensive picture by capturing changes with respect to all variables involved.

• Comprehensive Change: Unlike partial derivatives, which focus on change with respect to one variable, the total derivative accounts for simultaneous changes in all variables.
• Mathematical Representation: For a function \( F(x, y, z(x, y)) = 0 \), the total derivative with respect to x can be expressed as \( \frac{dF}{dx} = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial z} \cdot \frac{\partial z}{\partial x} = 0 \).
• Importance: Understanding total derivatives is crucial for solving problems where several variables influence a situation and change together, such as in implicit differentiation or systems modeling.

With the total derivative, mathematicians and scientists can efficiently predict how a system will behave when more than one factor is at play, making it an invaluable tool in fields ranging from physics to economics.