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Chapter 11: Problem 111

Suppose \(g(x)=-f(x)\). How do the derivatives of \(f\) and \(g\) compare?

Short Answer

Expert verified
The derivatives of functions f(x) and g(x) compare as follows: the derivative of g(x), denoted as g'(x), is the negative of the derivative of f(x), denoted as f'(x). Mathematically, this relationship is represented as \(g'(x) = -f'(x)\). This means that the slopes of the tangent lines to f(x) and g(x) at any point x are opposite in sign, but have the same magnitude.

Step by step solution

01

Find the derivative of f(x)

To find the derivative of f(x) with respect to x, we denote it as f'(x) or \(\frac{d}{dx}(f(x))\).
02

Find the derivative of g(x)

Since g(x) is equal to -f(x), to find the derivative of g(x) with respect to x, we denote it as g'(x) or \(\frac{d}{dx}(g(x))\). Using the chain rule of differentiation, we have: \(g'(x) = \frac{d}{dx}(-f(x))\) Now, simplify the equation: \(g'(x) = -\frac{d}{dx}(f(x))\)
03

Compare the derivatives of f and g

As we found in step 1, the derivative of f(x) is: f'(x) And, the derivative of g(x), as we found in step 2, is: g'(x) = \(-f'(x)\) This indicates that the derivative of g(x) is simply the negative of the derivative of f(x). In other words, the slopes of the tangent lines to f(x) and g(x) at any point x are opposite in sign, but have the same magnitude.

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Key Concepts

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

Differentiation
Differentiation is a fundamental concept in calculus that involves finding the rate at which a function changes at any point. This process gives us what is known as a derivative. Differentiating a function provides a way to determine how the function's output value changes with respect to changes in its input value. It is essential for understanding the behavior of functions.
  • When you differentiate a function, you seek its derivative, often represented as \(f'(x)\) for a function \(f(x)\).
  • This derivative is the instantaneous rate of change of the function, analogous to the slope of a tangent line at any point \(x\).
In our exercise, we are comparing the derivatives of two functions: \(f(x)\) and \(g(x) = -f(x)\). Differentiation allows us to calculate how these two functions change and compare their rates of change.
Chain Rule
The Chain Rule is a crucial method used in calculus for differentiating composite functions, which are functions made up of other functions. It is especially useful when dealing with combinations like \(-f(x)\), where one function is multiplied by a constant.
  • The Chain Rule states: if you have a function \(h(x) = f(g(x))\), then its derivative is \(h'(x) = f'(g(x)) \cdot g'(x)\).
  • In the context of differentiating our function \(g(x) = -f(x)\), the chain rule simplifies the process of finding \(g'(x)\). Here, it is applied to show that multiplying by a constant affects the derivative.
When applying the Chain Rule to \(-f(x)\), we observe that differentiating \(-1\) times a function results in \(-1\) times the derivative of the function, hence \(g'(x) = -f'(x)\).
Derivatives
Derivatives are at the core of calculus, representing the rate of change of a function in relation to its variable. Essentially, the derivative tells us the rate of change of the y-value (output) concerning a change in the x-value (input). This is fundamental in constructing an understanding of how functions behave.
  • Mathematically, the derivative of a function \(f(x)\) is noted by \(f'(x)\) or \(\frac{d}{dx}(f(x))\).
  • It describes the slope of the tangent line to the function at any given point, which correlates to the function's growth, decline, or constancy at that point.
From our exercise, derivatives help us identify the relationship between two functions, \(f(x)\) and its opposite, \(g(x) = -f(x)\). The process unveils that the derivative of \(g(x)\) is simply the negative of \(f'(x)\), emphasizing that changes in \(g(x)\) occur in a reverse manner compared to \(f(x)\).
Function Properties
Understanding function properties is vital to analyzing how functions are linked and how operations like negation affect them. When a function \(f(x)\) is manipulated, its properties, including derivatives, reflect these changes.
  • A function's properties include aspects like continuity, differentiability, and linearity.
  • In this context, knowing that \(g(x) = -f(x)\) implies symmetry about the x-axis due to the negative sign, offers an insight into how reflections impact the derivative.
By using the properties of functions, we can deduce that the derivatives of negative functions like \(g(x) = -f(x)\) will mirror the original function's rate of change in the opposite direction. As illustrated in the exercise, for any function \(f(x)\), if you flip the function over the x-axis to form \(g(x)\), each derivative is negatively scaled, \(g'(x) = -f'(x)\), showcasing this mirrored behavior in derivatives, crucial in calculus analysis.

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Most popular questions from this chapter

Calculate the derivatives of the functions in Exercises 1-46. HINT [See Example 1.] \(h(x)=\frac{1}{\left(x^{2}+x+1\right)^{2}}\)

Existing Home Sales The following graph shows the approximate value of home prices and existing home sales in 2004-2007 as a percentage change from 2003 , together with quadratic approximations.The quadratic approximations are given by Home Prices: $$ P(t)=-6 t^{2}+27 t+10(0 \leq t \leq 3) $$ Existing Home Sales: \(S(t)=-4 t^{2}+4 t+11 \quad(0 \leq t \leq 3)\) where \(t\) is time in years since the start of 2004 . Use the chain rule to estimate \(\left.\frac{d S}{d P}\right|_{t=2} .\) What does the answer tell you about home sales and prices? HINT [See Quick Example 2 on page 828.]

You and I are both selling a steady 20 T-shirts per day. The price I am getting for my T-shirts is increasing twice as fast as yours, but your T-shirts are currently selling for twice the price of mine. Whose revenue is increasing faster: yours, mine, or neither? Explain.

Calculate the derivatives of the functions in Exercises 1-46. HINT [See Example 1.] \(r(x)=\left(0.1 x-4.2 x^{-1}\right)^{0.5}\)

Use the calculation thought experiment to say whether the expression is written as a sum, difference, scalar multiple, product, or quotient. Then use the appropriate rules to find its derivative. HINT [See Quick Examples on page 814 and Example 7.] \(y=\frac{(x+2) x}{x+1}\) (Do not simplify the answer.)
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