Problem 45 - Calculus Early Transcendentals

Question

Use implicit differentiation to find \(\partial z / \partial x\) and \(\partial z / \partial y\) $$x^{2}+y^{2}+z^{2}=3 x y z$$

Step-by-Step Solution

Answer

\(\frac{\partial z}{\partial x} = \frac{3yz - 2x}{2z - 3xy}\) and \(\frac{\partial z}{\partial y} = \frac{3xz - 2y}{2z - 3xy}\).

Step 1: Rewriting the Function

Start with the given equation: \(x^2 + y^2 + z^2 = 3xyz\). This is the function we will differentiate implicitly.

Step 2: Implicit Differentiation w.r.t. x

Differentiate each term of the equation with respect to \(x\), treating \(y\) and \(z\) as functions of \(x\). This gives: \(2x + 2z \frac{dz}{dx} = 3(yz + xy \frac{dz}{dx})\).Rearrange to find \(\frac{\partial z}{\partial x}\).

Step 3: Solving for \(\frac{\partial z}{\partial x}\)

Rearrange the differentiated equation:\(2z \frac{dz}{dx} - 3xy \frac{dz}{dx} = 3yz - 2x\).Factor out \(\frac{dz}{dx}\):\(\frac{dz}{dx} (2z - 3xy) = 3yz - 2x\).Solve for \(\frac{dz}{dx}\):\(\frac{\partial z}{\partial x} = \frac{3yz - 2x}{2z - 3xy}\).

Step 4: Implicit Differentiation w.r.t. y

Differentiate each term of the equation with respect to \(y\), treating \(x\) and \(z\) as functions of \(y\). This gives: \(2y + 2z \frac{dz}{dy} = 3(xz + xy \frac{dz}{dy})\).Rearrange to find \(\frac{\partial z}{\partial y}\).

Step 5: Solving for \(\frac{\partial z}{\partial y}\)

Rearrange the differentiated equation:\(2z \frac{dz}{dy} - 3xy \frac{dz}{dy} = 3xz - 2y\).Factor out \(\frac{dz}{dy}\): \(\frac{dz}{dy}(2z - 3xy) = 3xz - 2y\).Solve for \(\frac{dz}{dy}\):\(\frac{\partial z}{\partial y} = \frac{3xz - 2y}{2z - 3xy}\).

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus. They express how a function changes as its variables are varied individually. For example, if you have a function involving variables \(x, y,\) and \(z\), the partial derivative with respect to \(x\) shows how the function's value changes as \(x\) alone changes, while \(y\) and \(z\) are held constant. Similarly, partial derivatives with respect to \(y\) and \(z\) reveal changes in those directions. In the given problem, you have to find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\). These represent the rates of change of \(z\) with respect to \(x\) and \(y\), respectively, while considering \(z\) as a dependent function of \(x\) and \(y\). Calculating these derivatives involves implicit differentiation, a technique that simplifies finding derivatives of functions defined implicitly by equations.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions of several variables. It's crucial when dealing with real-world scenarios where phenomena are dependent on multiple factors. Instead of just one variable affecting the outcome, multivariable calculus lets us understand the interplay between multiple inputs. In this context, the function \(x^2 + y^2 + z^2 = 3xyz\) is defined by three variables: \(x, y,\) and \(z\). The problem asks for derivatives that consider how one variable affects another in this multivariate environment. Solutions in this realm require techniques like implicit differentiation to handle the complexity of relationships among several variables.

Differentiation Techniques

Differentiation, the process of finding a derivative, can become intricate with multiple variables. Having a toolbox of techniques enhances your ability to tackle diverse problems. One of these techniques, featured in this problem, is implicit differentiation. Implicit differentiation is used when it is difficult to solve one variable explicitly in terms of others. Instead, we derive each side of the given equation directly with respect to the variable of interest, considering other variables as implicit functions. This allows you to isolate and differentiate terms accurately, and is necessary when working with equations like \(x^2 + y^2 + z^2 = 3xyz\) where variables are entangled. By understanding and applying implicit differentiation, you can successfully solve for partial derivatives, unveiling the hidden relationships between variables in complex equations.