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Chapter 3: Problem 7

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. $$\begin{array}{cccccc} x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & 2 & 3 & 4 & 6 & 7 \\ f^{\prime}(x) & 1 & 1 / 2 & 2 & 3 / 2 & 1 \end{array}$$ $$\text { a. }\left(f^{-1}\right)^{\prime}(4) \quad \text { b. }\left(f^{-1}\right)^{\prime}(6) \quad \text { c. }\left(f^{-1}\right)^{\prime}(1) \quad \text { d. } f^{\prime}(1)$$

Short Answer

Expert verified
Question: Find the derivatives of the inverse function, (f^{-1})'(4), (f^{-1})'(6), (f^{-1})'(1), and the derivative of the function f'(1). Answer: The respective derivatives are (f^{-1})'(4) = 1/2, (f^{-1})'(6) = 2/3, (f^{-1})'(1) is indeterminable, and f'(1) = 3/2.

Step by step solution

01

a. \((f^{-1})'(4)\)

To find \((f^{-1})'(4)\): 1. Determine \(f^{-1}(4)\) by looking at the \(f(x)\) table. We see that \(f(0) = 4\), so \(f^{-1}(4) = 0\). 2. Check the \(f'(x)\) table to find the derivative at this point: \(f'(0) = 2\). 3. Apply the formula for the derivative of the inverse function: \((f^{-1})'(4) = \frac{1}{f'(f^{-1}(4))}\), which in this case is \((f^{-1})'(4) = \frac{1}{2}\).
02

b. \((f^{-1})'(6)\)

To find \((f^{-1})'(6)\): 1. Determine \(f^{-1}(6)\) by looking at the \(f(x)\) table. We see that \(f(1) = 6\), so \(f^{-1}(6) = 1\). 2. Check the \(f'(x)\) table to find the derivative at this point: \(f'(1) = 3/2\). 3. Apply the formula for the derivative of the inverse function: \((f^{-1})'(6) = \frac{1}{f'(f^{-1}(6))}\), which in this case is \((f^{-1})'(6) = \frac{1}{3/2} = \frac{2}{3}\).
03

c. \((f^{-1})'(1)\)

To find \((f^{-1})'(1)\): 1. Determine \(f^{-1}(1)\) by looking at the \(f(x)\) table. However, we see that there is no \(x\) value that has \(f(x) = 1\). As a result, we cannot determine \((f^{-1})'(1)\).
04

d. \(f'(1)\)

We are directly asked for the derivative of the function \(f(x)\) at \(x=1\). Check the \(f'(x)\) table to find the derivative at this point: \(f'(1) = 3/2\). The respective derivatives are \((f^{-1})'(4) = \frac{1}{2}\), \((f^{-1})'(6) = \frac{2}{3}\), \((f^{-1})'(1)\) is indeterminable, and \(f'(1) = \frac{3}{2}\).

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Key Concepts

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

Inverse Function Calculus
Inverse functions play a key role in calculus by allowing us to switch perspectives between the inputs and outputs of a given function. Inverse function calculus is particularly pivotal when we want to find the rate of change of the inverse function at a certain point. To get the derivative of an inverse function, we use a special formula: If we have a function f and its inverse f-1, and if both functions are differentiable, then the derivative of the inverse function at a point y is given by (f-1)'(y) = 1 / f'(f-1(y)), provided f'(f-1(y)) is not zero.

This formula is rooted in the chain rule of derivatives and reflects the interrelated slopes of the function and its inverse. Understanding this relationship helps students solve problems involving derivatives of inverse functions without necessarily having the explicit form of the inverse function. When dealing with such problems, it's important to locate the corresponding value in the function's range to find the derivative at the inverse's domain value.
Table of Values Derivatives
A table of values can provide a convenient way to visually inspect the behavior of a function and its derivative. When dealing with derivatives from a table of values, we can follow a routine methodology. To extract information about derivatives, one should:
  • Identify the function f(x) and its corresponding derivative f'(x) from the table.
  • Match the y value for which the derivative is needed with the correct x value using the function's information.
  • Locate the derivative f'(x) for that specific x value.
For inverse functions, the process involves an additional step of finding the x value for which f(x) equals the given y value, then using this x value in the derivative formula as explained in the previous section.

It's crucial to interpret the table correctly because if the y value isn't present in the function's range, as revealed in the table, we can't find the inverse's derivative at that y value, as seen in the exercise's part c. Thus, attention to detail and careful matching are essential skills when working with such tables.
Derivative at a Point
The derivative of a function at a specific point provides the slope of the tangent line to the function's graph at that point. It represents the instantaneous rate of change of the function with respect to its input at that very moment. When you have a table of values, as we do in our textbook exercise, finding the derivative at a point is as simple as identifying the corresponding f'(x) value for your chosen x value.

In practice, you might also be asked to calculate the derivative at a point for an inverse function, which requires a bit more work involving the original function and its derivative, as you need to first find the inverse function's value that maps to the given point. Remember, the critical piece to note is that the derivative at a point is not always available directly and might involve an additional step of computation, particularly when one is working with inverse functions or when the function鈥檚 expression is not directly provided.

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