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Chapter 14: Problem 55

Which order of differentiation will calculate \(f_{x y}\) faster: \(x\) first or \(y\) first? Try to answer without writing anything down. $$\begin{array}{l}{\text { a. } f(x, y)=x \sin y+e^{y}} \\ {\text { b. } f(x, y)=1 / x} \\ {\text { c. } f(x, y)=y+(x / y)} \\ {\text { d. } f(x, y)=y+x^{2} y+4 y^{3}-\ln \left(y^{2}+1\right)} \\ {\text { e. } f(x, y)=x^{2}+5 x y+\sin x+7 e^{x}} \\ {\text { f. } f(x, y)=x \ln x y}\end{array}$$

Short Answer

Expert verified
For (a) either equally, (b) y first, (c) x first, (d) y first, (e) y first, (f) y first.

Step by step solution

01

Understanding the Problem

We need to calculate the mixed partial derivatives \( f_{xy} \) and \( f_{yx} \) for each given function \( f(x, y) \) and determine which order鈥攄ifferentiating with respect to \( x \) first or \( y \) first鈥攊s faster for each sub-problem. "Faster" means requiring less simplification or having more straightforward derivative rules when evaluating.
02

Function Analysis: Part a

For \( f(x, y) = x \sin y + e^y \), the order of differentiation doesn't affect the speed significantly as both partial derivatives involve similar straightforward rules. Both partial derivatives require simple derivatives of basic functions. Thus, either order (\( x \) first or \( y \) first) will yield the same effort in computation.
03

Function Analysis: Part b

For \( f(x, y) = 1/x \), differentiating with respect to \( y \) results in zero since \( 1/x \) is independent of \( y \). Therefore, calculating \( f_{xy} \) involves differentiating \( 0 \) with respect to \( x \), which is trivially zero. Thus, it's faster to differentiate with respect to \( y \) first.
04

Function Analysis: Part c

For \( f(x, y)=y+(x/y) \), differentiating first with respect to \( x \) gives \( 1/y \). Then differentiating \( 1/y \) with respect to \( y \) is simpler (\(-1/y^2\)) than starting with differentiation with respect to \( y \), which involves quotient rule initially. Hence, \( x \) first is faster.
05

Function Analysis: Part d

For \( f(x, y) = y + x^2 y + 4y^3 - \ln(y^2 + 1) \), differentiating with respect to \( y \) simplifies the expression quickly as many terms have \( y \) in them. Thus, differentiation with respect to \( y \) first reduces complexity initially compared to \( x \) first because \( f_y \) directly addresses these terms.
06

Function Analysis: Part e

For \( f(x, y) = x^2 + 5xy + \sin x + 7 e^x \), it's faster to differentiate with respect to \( y \) first. Many terms are independent of \( y \) and thus make the partial derivative \( f_y \) simpler and quicker.
07

Function Analysis: Part f

For \( f(x, y) = x \ln(xy) \), differentiating with respect to \( y \) first is simpler due to the logarithmic differentiation rules. Differential rules for \( \ln(xy) \) resolve more directly with \( y \) initial differentiation.
08

Conclusion

Evaluate the order of differentiation based on the complexity of the partial derivatives for each specific function. In easy cases where the function simplifies significantly, that order should be preferred.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mixed Partial Derivatives
Mixed partial derivatives are tools used to analyze the behavior of functions with more than one variable. When dealing with a function of two variables, say \( f(x, y) \), the mixed partial derivative \( f_{xy} \) represents the rate at which the derivative of \( f \) with respect to \( x \) changes as \( y \) changes. Similarly, \( f_{yx} \) involves differentiating first with respect to \( y \) and then \( x \).To calculate a mixed partial derivative, you follow these steps:
  • Differentiating the function partially, first with respect to one variable.
  • Differentiating the resulting expression partially with respect to the other variable.
These derivatives are used extensively in function analysis, economics, physics, and engineering to understand the interplay between variables.The equality \( f_{xy} = f_{yx} \), as stated by Schwarz's theorem (given the function is continuous and well-behaved), tells us that the order of differentiation at a given point does not affect the final result. However, the path to the result might be easier or quicker depending on the function's structure.
Order of Differentiation
The order of differentiation refers to the sequence in which you apply partial derivatives to a function. For example, in the problem, differentiating a function \( f(x, y) \) consecutively in different orders produces two potential results: \( f_{xy} \) and \( f_{yx} \).A key aspect is to identify which order simplifies the calculation process. This depends greatly on the nature of the function:
  • If the function has terms where one variable is not present (e.g., independent of \( y \) in \( f(x, y) = 1/x \)), differentiating with respect to that variable first can lead to quicker simplification.
  • Functions involving products or compositions of different types of functions (e.g., polynomial, logarithmic, trigonometric) may simplify differently depending on which variable you differentiate first.
Selecting the right order can reduce computational efforts, especially in complex expressions or when applying these derivatives in more extensive analyses.
Function Analysis
Function analysis in this context involves evaluating how a function's structure affects the process of finding its derivatives. Each function in the exercise, such as \( f(x, y) = y + (x/y) \), showcases various mathematical features.To decide the optimal order of differentiation:
  • Identify the main operations involved within the function (addition, multiplication, logarithms, etc.).
  • Assess whether simplifying or breaking down these operations beforehand can make the differentiation steps easier to perform.
  • Consider function components that are independent of either variable to potentially expedite simplification cycles.
This assessment inherently improves familiarity with function classes, influencing quicker determination of the mixed partial derivatives.
Simplification of Derivatives
Simplifying derivatives makes the analytical process more efficient. This involves using mathematical properties and rules to condense the expressions obtained after differentiation. Some strategies to simplify derivatives include:
  • Applying derivative rules carefully, such as the product, quotient, and chain rules, to effectively reduce complexity.
  • Analyzing terms individually, to identify simplifications from identities or constants.
  • Substituting expressions where necessary to convert difficult components to simpler equivalents.
The aim in simplification is minimizing the manipulations and computations required, especially vital in higher-level calculus where functions can be large and intricate. By thoughtfully structuring differentiation processes, regardless of initial complexity, these tasks can be approached with organized and logical steps.

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Most popular questions from this chapter

You will explore functions to identify their local extrema. Use a CAS to perform the following steps: a. Plot the function over the given rectangle. b. Plot some level curves in the rectangle. c. Calculate the function's first partial derivatives and use the CAS equation solver to find the critical points. How do the critical points relate to the level curves plotted in part (b)? Which critical points, if any, appear to give a saddle point? Give reasons for your answer. d. Calculate the function's second partial derivatives and find the discriminant \(f_{x x} f_{y y}-f_{x y}^{2}\) . e. Using the max-min tests, classify the critical points found in part (c). Are your findings consistent with your discussion in part (c)? $$f(x, y)=\left\\{\begin{array}{ll}{x^{5} \ln \left(x^{2}+y^{2}\right),} & {(x, y) \neq(0,0)} \\ {0,} & {(x, y)=(0,0)}\end{array}\right.$$ $$-2 \leq x \leq 2, \quad-2 \leq y \leq 2$$

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Find the value of \(\partial x / \partial z\) at the point \((1,-1,-3)\) if the equation $$x z+y \ln x-x^{2}+4=0$$ defines \(x\) as a function of the two independent variables \(y\) and \(z\) and the partial derivative exists.

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