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Chapter 14: Problem 5

Find the partial derivatives. The variables are restricted to a domain on which the function is defined. $$\frac{\partial z}{\partial x} \text { and } \frac{\partial z}{\partial y} \text { if } z=\left(x^{2}+x-y\right)^{7}$$

Short Answer

Expert verified
\( \frac{\partial z}{\partial x} = 7(2x + 1)(x^2 + x - y)^6 \) and \( \frac{\partial z}{\partial y} = -7(x^2 + x - y)^6 \).

Step by step solution

01

Understand the Problem

We need to find the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \) of the function \( z = (x^2 + x - y)^7 \). A partial derivative measures how a function changes as only one of the independent variables change while the other variables are held constant.
02

Apply the Chain Rule for \( \frac{\partial z}{\partial x} \)

To find \( \frac{\partial z}{\partial x} \), apply the chain rule. Let \( u = x^2 + x - y \), so we have \( z = u^7 \).1. Find \( \frac{\partial z}{\partial u} = 7u^6 \).2. Then find \( \frac{\partial u}{\partial x} = 2x + 1 \). 3. Use the chain rule: \( \frac{\partial z}{\partial x} = \frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial x} = 7(x^2 + x - y)^6 \cdot (2x + 1) \).
03

Simplify \( \frac{\partial z}{\partial x} \)

Simplify the expression obtained: \[ \frac{\partial z}{\partial x} = 7(2x + 1)(x^2 + x - y)^6 \].
04

Apply the Chain Rule for \( \frac{\partial z}{\partial y} \)

Next, find \( \frac{\partial z}{\partial y} \) using a similar process. 1. We already have \( \frac{\partial z}{\partial u} = 7u^6 \).2. Find \( \frac{\partial u}{\partial y} = -1 \) since \( u = x^2 + x - y \).3. Use the chain rule: \( \frac{\partial z}{\partial y} = \frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial y} = 7(x^2 + x - y)^6 \cdot (-1) \).
05

Simplify \( \frac{\partial z}{\partial y} \)

Simplify the expression obtained: \[ \frac{\partial z}{\partial y} = -7(x^2 + x - y)^6 \].
06

Conclusion

The partial derivatives are:1. \( \frac{\partial z}{\partial x} = 7(2x + 1)(x^2 + x - y)^6 \).2. \( \frac{\partial z}{\partial y} = -7(x^2 + x - y)^6 \).

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Key Concepts

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

Chain Rule
When dealing with functions of multiple variables, like our exercise function \( z = (x^2 + x - y)^7 \), the chain rule helps simplify the differentiation process. The chain rule is used when a variable, say \( z \), is a function of another variable, \( u \), which is itself a function of yet another variable. This is perfect when we have composite functions.

In the case of finding \( \frac{\partial z}{\partial x} \), we first set \( u = x^2 + x - y \), turning our function into \( z = u^7 \).
  • First, find \( \frac{\partial z}{\partial u} \) by differentiating \( z = u^7 \) with respect to \( u \), resulting in \( 7u^6 \).
  • Next, find \( \frac{\partial u}{\partial x} \) by differentiating \( x^2 + x - y \) with respect to \( x \), giving \( 2x + 1 \).
  • Finally, apply the product of these derivatives to find \( \frac{\partial z}{\partial x} = 7(x^2 + x - y)^6(2x + 1) \).
The chain rule simplifies complex derivative calculations by efficiently breaking them down into simpler parts.
Multivariable Calculus
Multivariable calculus involves dealing with functions that depend on more than one variable. Unlike single-variable calculus where functions depend on one variable, multivariable calculus expands this idea, helping us describe a vast range of real-world situations like physics and engineering problems.

In our exercise, \( z = (x^2 + x - y)^7 \) involves three variables \( x \), \( y \), and \( z \). To analyze how \( z \) changes, we need to determine the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).
  • Partial derivatives provide us a way to observe how a small change in one variable affects the overall function while holding other variables constant.
  • This approach helps in understanding the behavior of complex systems modeled by multivariable functions.
  • It offers insights into how different components of a system influence each other.
Overall, multivariable calculus enhances our ability to solve and analyze complicated mathematical models.
Function Differentiation
Differentiation of functions, especially in calculus, is the process of finding the derivative, which is a measure of how a function changes as its input changes.

For our given function, \( z = (x^2 + x - y)^7 \), differentiation helps in understanding how the output \( z \) reacts to changes in \( x \) or \( y \).
  • The process involves using rules and formulas, like the chain rule, to simplify and calculate derivatives efficiently.
  • By finding \( \frac{\partial z}{\partial x} \), it indicates how \( z \) changes with a small change in \( x \).
  • Similarly, \( \frac{\partial z}{\partial y} \) shows how \( z \) alters with a small change in \( y \).
  • These partial derivatives give the rates of change in the direction of each variable separately.
Mastering function differentiation is key to solving real-world problems involving dynamic systems with varying inputs and conditions.

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Most popular questions from this chapter

Are the statements true or false? Give measons for your answer. If \(f(x, y)\) is a constant function, then \(d f=0\)

True or false? Give reasons for your answer. At the point \((3,0),\) the function \(g(x, y)=x^{2}+y^{2}\) has the same maximal rate of increase as that of the function \(h(x, y)=2 x y\)

Find the gradient of the function. Assume the variables are restricted to a domain on which the function s defined. $$f(x, y)=\ln \left(x^{2}+y^{2}\right)$$

For the differentiable function \(h(x, y),\) we are told that \(h(600,100)=300\) and \(h_{x}(600,100)=12\) and \(h_{y}(600,100)=-8 .\) Estimate \(h(605,98)\)

Explain what is wrong with the statement. There is a function \(f(x, y)\) with partial derivatives \(f_{x}=\) \(x y\) and \(f_{y}=y^{2}\)
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