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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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Inverse Cosine

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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Related Practice
Textbook Question

{Use of Tech} Power and energy Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up. It is measured in units of joules or Calories, where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶J, or 240 Cal. On the other hand, power is the rate at which energy is used, which is measured in watts, where 1 W=1 J/s. Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for one hour, the total amount of energy used is 1 kilowatt-hour (1 kWh=3.6×10⁶ J) Suppose the cumulative energy used in a large building over a 24-hr period is given by E(t)=100t+4t²−t³ / 9kWh where t=0 corresponds to midnight.

c. Graph the power function and interpret the graph. What are the units of power in this case?

Textbook Question

{Use of Tech} Spring oscillations A spring hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance x (in inches) of the mass from its equilibrium position after t seconds is given by the function x(t) = 10sin t−10cos t, where x is positive when the mass is above the equilibrium position. <IMAGE>

c. At what times is the velocity of the mass zero?

Textbook Question

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

c. d/dx (sin⁴ x)

Textbook Question

Computing the derivative of f(x) = e^-x

c. Use parts (a) and (b) to find the derivative of f(x) = e^-x.

Textbook Question

Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.

73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.

c. f(x) = √|x-4|

Textbook Question

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>


{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.


c. How fast (in fish per year) is the population growing at t=0? At t=5?

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