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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 3.7.25d
Chapter 3, Problem 3.7.25d

Derivatives using tables Let h(x)=f(g(x))h(x)=f(g(x)) and p(x)=g(f(x))p(x)=g(f(x)). Use the table to compute the following derivatives.
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d. p′(2)p^{\(\prime\)}\(\left\)(2\(\right\))

Verified step by step guidance
1
Identify that you need to find the derivative of the composite function p(x) = g(f(x)) at x = 2, which is p'(2).
Recall the chain rule for derivatives, which states that if you have a composite function p(x) = g(f(x)), then the derivative p'(x) = g'(f(x)) * f'(x).
Evaluate f(x) at x = 2 using the table to find f(2). This will give you the input for g'.
Use the table to find g'(f(2)), which is the derivative of g at the point f(2).
Find f'(2) using the table, which is the derivative of f at x = 2. Multiply g'(f(2)) by f'(2) to get p'(2).

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Key Concepts

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

Chain Rule

The Chain Rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function h(x) is composed of two functions f and g, such that h(x) = f(g(x)), then the derivative h'(x) can be found using the formula h'(x) = f'(g(x)) * g'(x). This rule is essential for calculating derivatives of functions that are nested within each other.
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Intro to the Chain Rule

Derivative Notation

Derivative notation, such as f'(x) or p'(2), represents the rate of change of a function with respect to its variable. The notation p'(2) specifically indicates the derivative of the function p evaluated at the point x = 2. Understanding this notation is crucial for interpreting and calculating derivatives accurately.
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Sigma Notation

Function Composition

Function composition occurs when one function is applied to the result of another function. In the context of the question, h(x) = f(g(x)) and p(x) = g(f(x)) are examples of composed functions. Recognizing how to work with composed functions is vital for applying the Chain Rule and finding derivatives of such functions.
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Evaluate Composite Functions - Special Cases
Related Practice
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.


d. Graph P' and use the graph to estimate the year in which the population is growing fastest. 

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Textbook Question

Airline travel The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>

d. Determine the velocity of the airliner at noon (t = 6) and explain why the velocity is negative.

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Textbook Question

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>

d. f'(1)

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Textbook Question

Derivatives using tables Let h(x)=f(g(x))h(x)=f(g(x)) and p(x)=g(f(x))p(x)=g(f(x)). Use the table to compute the following derivatives.

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c. p′(4)p^{\(\prime\)}\(\left\)(4\(\right\))

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Textbook Question

Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.

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

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Textbook Question

A rectangular swimming pool 10 ft wide by 20 ft long and of uniform depth is being filled with water.

c. At what rate is the water level rising if the pool is filled at a rate of 10ft³/min?

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