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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.10.7c
Chapter 3, Problem 3.10.7c

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>
c. (f^-1)'(1)

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Identify the relationship between a function and its inverse. If \( y = f(x) \), then \( x = f^{-1}(y) \). The derivative of the inverse function \( (f^{-1})'(y) \) can be found using the formula \( (f^{-1})'(y) = \frac{1}{f'(x)} \) where \( x = f^{-1}(y) \).
From the problem, we need to find \( (f^{-1})'(1) \). This means we need to find the value of \( x \) such that \( f(x) = 1 \).
Look at the table provided in the problem to find the value of \( x \) for which \( f(x) = 1 \). This will give us the point \( (x, 1) \) on the graph of \( f \).
Once the correct \( x \) is identified, use the table to find \( f'(x) \), the derivative of \( f \) at this \( x \).
Finally, apply the formula \( (f^{-1})'(1) = \frac{1}{f'(x)} \) using the value of \( f'(x) \) obtained from the table to find the derivative of the inverse function at the given point.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Functions

Inverse functions are functions that 'reverse' the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f^-1(y) takes y back to x. Understanding how to find and work with inverse functions is crucial for determining their derivatives.
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Derivative of Inverse Functions

The derivative of an inverse function can be calculated using the formula (f^-1)'(y) = 1 / f'(x), where y = f(x). This relationship shows that the derivative of the inverse function at a point is the reciprocal of the derivative of the original function at the corresponding point. This concept is essential for solving problems involving derivatives of inverse functions.
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Derivatives of Inverse Sine & Inverse Cosine

Using Tables for Derivatives

When working with derivatives from tables, it is important to locate the necessary values for the function and its derivative. The table typically provides values of f(x) and f'(x) at specific points, which can be used to find the derivative of the inverse function. Understanding how to interpret and extract information from these tables is key to solving derivative problems.
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Derivatives
Related Practice
Textbook Question

{Use of Tech} Spring runoff The flow of a small stream is monitored for 90 days between May 1 and August 1. The total water that flows past a gauging station is given by v(t) = <matrix 2x2> where V is measured in cubic feet and t is measured in days, with t=0 corresponding to May 1.

c. Describe the flow of the stream over the 3-month period. Specifically, when is the flow rate a maximum?

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Textbook Question

Another way to approximate derivatives is to use the centered difference quotient: f' (a) ≈ f(a+h) - f(a- h) / 2h. Again, consider f(x) = √x.

c. Explain why it is not necessary to use negative values of h in the table of part (b).

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Textbook Question

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x)=(sec^−1 x)/x on [1,∞)

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Textbook Question

Derivatives of sin^n x Calculate the following derivatives using the Product Rule.

c. d/dx (sin⁴ x)

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Textbook Question

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 19.6 m/s from a height of 24.5 m above the ground. The height (in meters) of the stone above the ground t seconds after it is thrown is s(t) = -4.9t²+19.6t+24.5.

c. What is the height of the stone at the highest point?

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Textbook Question

Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position y=0y=0 when the mass hangs at rest. Suppose you push the mass to a position y0y_0 units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is y=y0cos(tkm)y=y_0\(\cos\]\left\)(t\(\sqrt{\frac{k}{m}\)}\(\right\)), where k>0k>0 is a constant measuring the stiffness of the spring (the larger the value of kk, the stiffer the spring) and yy is positive in the upward direction.

Use equation (4) to answer the following questions.

c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness (kk is increased by a factor of 44)?

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