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Magazine Chapter 12: Problem 18
If \(F(u, v)=\tan ^{-1}(u v), u=\sqrt{x y},\) and \(v=\sqrt{x}-\sqrt{y}\) find \(\partial F / \partial x\) and \(\partial F / \partial y\) in terms of \(u, v, x,\) and \(y\)
Short Answer
Expert verified
Use the chain rule and simplify for \(\frac{\partial F}{\partial x}\) and \(\frac{\partial F}{\partial y}\).
Step by step solution
01
Understand the problem
We are given a function \(F(u, v) = \tan^{-1}(uv)\) and two substitutions, \(u = \sqrt{xy}\) and \(v = \sqrt{x} - \sqrt{y}\). We need to find the partial derivatives \(\frac{\partial F}{\partial x}\) and \(\frac{\partial F}{\partial y}\) with respect to \(x\) and \(y\), respectively.
02
Compute partial derivatives of F with respect to u and v
Using the chain rule, the partial derivative of \(F\) with respect to \(u\) is \(\frac{\partial F}{\partial u} = \frac{1}{1 + (uv)^2} v\), and with respect to \(v\) is \(\frac{\partial F}{\partial v} = \frac{1}{1 + (uv)^2} u\).
03
Use chain rule for \(\partial F / \partial x\)
Apply the chain rule: \(\frac{\partial F}{\partial x} = \frac{\partial F}{\partial u} \cdot \frac{\partial u}{\partial x} + \frac{\partial F}{\partial v} \cdot \frac{\partial v}{\partial x}\). For \(\frac{\partial u}{\partial x} = \frac{1}{2\sqrt{xy}} y\) and \(\frac{\partial v}{\partial x} = \frac{1}{2\sqrt{x}}\).
04
Calculate \(\partial F / \partial y\) using chain rule
Similar to the last step, \(\frac{\partial F}{\partial y} = \frac{\partial F}{\partial u} \cdot \frac{\partial u}{\partial y} + \frac{\partial F}{\partial v} \cdot \frac{\partial v}{\partial y}\). Here, \(\frac{\partial u}{\partial y} = \frac{1}{2\sqrt{xy}} x\) and \(\frac{\partial v}{\partial y} = -\frac{1}{2\sqrt{y}}\).
05
Substitute and simplify the expressions
Substitute all derivatives back into the equations: \[\frac{\partial F}{\partial x} = \frac{v}{1 + (uv)^2} \frac{y}{2\sqrt{xy}} + \frac{u}{1 + (uv)^2} \frac{1}{2\sqrt{x}}\] and \[\frac{\partial F}{\partial y} = \frac{v}{1 + (uv)^2} \frac{x}{2\sqrt{xy}} - \frac{u}{1 + (uv)^2} \frac{1}{2\sqrt{y}}\]. Simplify as much as possible.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chain Rule in Multivariable Calculus
The chain rule is a crucial tool in calculus, particularly when dealing with multivariable functions. It allows us to differentiate a function based on another function, or through several layers of inner functions.
In the given problem, we have a function \( F(u, v) = \tan^{-1}(uv) \) where \( u \) and \( v \) are themselves functions of \( x \) and \( y \).
The chain rule helps in understanding how small changes in \( x \) or \( y \) will affect \( F \) by first looking at their effects on \( u \) and \( v \).
This helps break down a complex problem into manageable parts by considering each variable's contribution independently.
In the given problem, we have a function \( F(u, v) = \tan^{-1}(uv) \) where \( u \) and \( v \) are themselves functions of \( x \) and \( y \).
The chain rule helps in understanding how small changes in \( x \) or \( y \) will affect \( F \) by first looking at their effects on \( u \) and \( v \).
- For partial derivatives like \( \frac{\partial F}{\partial x} \), the chain rule is expressed as:
\[\frac{\partial F}{\partial x} = \frac{\partial F}{\partial u} \cdot \frac{\partial u}{\partial x} + \frac{\partial F}{\partial v} \cdot \frac{\partial v}{\partial x}\] - Similarly, for \( \frac{\partial F}{\partial y} \):
\[\frac{\partial F}{\partial y} = \frac{\partial F}{\partial u} \cdot \frac{\partial u}{\partial y} + \frac{\partial F}{\partial v} \cdot \frac{\partial v}{\partial y}\]
This helps break down a complex problem into manageable parts by considering each variable's contribution independently.
Understanding Implicit Differentiation
Implicit differentiation becomes useful when functions are not given explicitly, but rather in a relationship form, such as \( u = \sqrt{xy} \) and \( v = \sqrt{x} - \sqrt{y} \).
Here, \( u \) and \( v \) are defined in terms of \( x \) and \( y \), instead of having a direct function relationship like \( u = f(x) \). This is where implicit differentiation comes into play.
This method bridges the gap when direct substitution isn't feasible, allowing us to navigate through interdependent variables.
Here, \( u \) and \( v \) are defined in terms of \( x \) and \( y \), instead of having a direct function relationship like \( u = f(x) \). This is where implicit differentiation comes into play.
- When calculating \( \frac{\partial u}{\partial x} \) or \( \frac{\partial v}{\partial x} \), we consider the derivatives of the respective expressions involving both \( x \) and \( y \). For example:
- \( \frac{\partial u}{\partial x} = \frac{1}{2\sqrt{xy}} y \) - Similarly, to find \( \frac{\partial v}{\partial y} \):
- \( \frac{\partial v}{\partial y} = -\frac{1}{2\sqrt{y}} \)
This method bridges the gap when direct substitution isn't feasible, allowing us to navigate through interdependent variables.
Exploring Multivariable Calculus Concepts
Multivariable calculus extends single-variable calculus concepts to functions of several variables.
In the problem, we explore functions of several variables: \( F(u, v) \) alongside \( u(x, y) \) and \( v(x, y) \).
This means instead of focusing on how a function changes with a single variable, we explore changes across multiple dimensions.
Understanding these concepts empowers us to tackle real-world problems involving complex systems with multiple changing variables.
In the problem, we explore functions of several variables: \( F(u, v) \) alongside \( u(x, y) \) and \( v(x, y) \).
This means instead of focusing on how a function changes with a single variable, we explore changes across multiple dimensions.
- Partial Derivatives: These are derivatives with respect to one variable while keeping the others constant, crucial in finding \( \frac{\partial F}{\partial x} \) and \( \frac{\partial F}{\partial y} \).
- The Notion of Tangency: At a specific point, partial derivatives resemble the slopes of tangent lines to surfaces, depicting the rate of change in specified directions.
- Applications: These principles are pivotal in engineering fields, economics, physical sciences, and more, where functions depend on several inputs.
Understanding these concepts empowers us to tackle real-world problems involving complex systems with multiple changing variables.
