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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 3.7.25c
Chapter 3, Problem 3.7.25c

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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1
Identify the function composition for p(x) = g(f(x)). We need to find the derivative p'(x) using the chain rule.
Apply the chain rule for derivatives: If p(x) = g(f(x)), then p'(x) = g'(f(x)) * f'(x).
Evaluate f(x) at x = 4 to find f(4). Use the table to find the value of f(4).
Use the table to find f'(4), the derivative of f at x = 4.
Substitute f(4) into g'(f(x)) to find g'(f(4)) using the table, then multiply g'(f(4)) by f'(4) to find p'(4).

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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'(4), represents the rate of change of a function at a specific point. The notation p'(4) indicates the derivative of the function p(x) evaluated at x = 4. Understanding this notation is crucial for interpreting the results of differentiation and applying them to specific values.
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Sigma Notation

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For example, evaluating p(4) means substituting 4 into the function p(x) to find its value. This concept is important when calculating derivatives, as it often requires evaluating the original functions at certain points to find the necessary derivatives for applying the Chain Rule.
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Evaluating Composed Functions
Related Practice
Textbook Question

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 64 ft/s from a height of 32 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+64t+32.

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

Textbook Question

Deriving trigonometric identities

c. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.

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?

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)?

Textbook Question

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

c. Solve the equation y(x²+4)=8 for y to find an explicit expression for y and then calculate dy/dx.