Textbook Question
Right circular cone The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation
______
S = πr √ r² + h².
b. How is dS/dt related to dh/dt if r is constant?
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.3.62b
The Reciprocal Rule
b. Show that the Reciprocal Rule and the Derivative Product Rule together imply the Derivative Quotient Rule.
Verified step by step guidance1
Start by recalling the Reciprocal Rule, which states that if you have a function \( f(x) \), the derivative of its reciprocal \( \frac{1}{f(x)} \) is given by \( -\frac{f'(x)}{(f(x))^2} \).
Next, recall the Product Rule for derivatives, which states that if you have two functions \( u(x) \) and \( v(x) \), the derivative of their product \( u(x)v(x) \) is \( u'(x)v(x) + u(x)v'(x) \).
To derive the Quotient Rule, consider a function \( \frac{u(x)}{v(x)} \). This can be rewritten as \( u(x) \cdot \frac{1}{v(x)} \).
Apply the Product Rule to \( u(x) \cdot \frac{1}{v(x)} \). Let \( u(x) \) be the first function and \( \frac{1}{v(x)} \) be the second function. The derivative is \( u'(x) \cdot \frac{1}{v(x)} + u(x) \cdot \left(-\frac{v'(x)}{(v(x))^2}\right) \).
Simplify the expression from the previous step to obtain the Quotient Rule: \( \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). This shows how the Reciprocal Rule and the Product Rule together imply the Quotient Rule.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reciprocal Rule
The Reciprocal Rule in calculus states that the derivative of the reciprocal of a function f(x), denoted as 1/f(x), is given by -f'(x)/[f(x)]^2. This rule is essential for understanding how changes in the reciprocal of a function relate to changes in the function itself, particularly when combined with other derivative rules.
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5:50
Power Rules
Derivative Product Rule
The Derivative Product Rule is a fundamental concept in calculus used to find the derivative of the product of two functions. If u(x) and v(x) are differentiable functions, the derivative of their product u(x)v(x) is given by u'(x)v(x) + u(x)v'(x). This rule is crucial for handling derivatives involving products of functions, which is often encountered in complex calculus problems.
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Derivative Quotient Rule
The Derivative Quotient Rule provides a method for differentiating a quotient of two functions. If u(x) and v(x) are differentiable functions, the derivative of their quotient u(x)/v(x) is given by [u'(x)v(x) - u(x)v'(x)]/[v(x)]^2. Understanding this rule is vital for solving problems involving division of functions, and it can be derived using the Reciprocal Rule and the Product Rule.
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Related Practice
Textbook Question
Analyzing Motion Using Graphs
[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:
b. When does it move to the left (down) or to the right (up)?
s = 200t - 16t², 0 ≤ t ≤ 12.5 (a heavy object fired straight up from Earth’s surface at 200 ft/sec)
Textbook Question
Particle motion At time t ≥ 0, the velocity of a body moving along the horizontal s-axis is v = t² − 4t + 3.
b. When is the body moving forward? Backward?
Textbook Question
Theory and Examples
In Exercises 51–54,
b. Graph y = f(x) and y = f'(x) side by side using separate sets of coordinate axes, and answer the following questions.
y = x⁴/4
Textbook Question
Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.
x ƒ(x) g(x) ƒ'(x) g'(x)
0 1 1 -3 1/2
1 3 5 1/2 -4
Find the first derivatives of the following combinations at the given value of x.
b. ƒ(x)g²(x), x = 0
Textbook Question
Recovering a function from its derivative
b. Repeat part (a), assuming that the graph starts at (−2, 0) instead of (−2, 3).