Textbook Question
Assume that y = 5x and dx/dt = 2. Find dy/dt
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 3.5.31
Derivatives
In Exercises 27–32, find dp/dq.
p = (q sin q) / (q² − 1)
Verified step by step guidance1
Step 1: Identify the function p = \(\frac{q \sin q}{q^2 - 1}\). This is a quotient of two functions, so we will use the Quotient Rule to find the derivative dp/dq.
Step 2: Recall the Quotient Rule, which states that if you have a function \(\frac{u}{v}\), its derivative is \(\frac{du/dq \cdot v - u \cdot dv/dq}{v^2}\). Here, u = q \(\sin\) q and v = q^2 - 1.
Step 3: Differentiate u = q \(\sin\) q with respect to q. Use the Product Rule: \(\frac{d}{dq}\)(q \(\sin\) q) = \(\frac{d}{dq}\)(q) \(\cdot\) \(\sin\) q + q \(\cdot\) \(\frac{d}{dq}\)(\(\sin\) q).
Step 4: Differentiate v = q^2 - 1 with respect to q. This is a simple power rule: \(\frac{d}{dq}\)(q^2 - 1) = 2q.
Step 5: Substitute the derivatives found in Steps 3 and 4 into the Quotient Rule formula from Step 2, and simplify the expression to find dp/dq.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate at which a function changes as its input changes. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The derivative is denoted as dp/dq, indicating the change in p with respect to the change in q.
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Derivatives
Quotient Rule
The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If p = f(q)/g(q), the derivative dp/dq is given by (g(q) * f'(q) - f(q) * g'(q)) / (g(q))². This rule is essential for differentiating functions like p in the given exercise, where p is expressed as a fraction.
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06:43The Quotient Rule
Trigonometric Functions
Trigonometric functions, such as sine and cosine, are fundamental in calculus and describe relationships between angles and sides of triangles. In the context of the given function p = (q sin q) / (q² − 1), the sine function introduces periodic behavior, which can affect the derivative's behavior. Understanding how to differentiate these functions is crucial for solving the problem.
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6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Derivatives
In Exercises 1–18, find dy/dx.
f(x) = x³ sin x cos x
Textbook Question
In Exercises 41–58, find dy/dt.
y = 4 sin(√(1 + √t))
Textbook Question
Finding Derivative Values
In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.
f(u) = 1 − (1/u), u = g(x) = (1 / (1 − x)), x = −1
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Textbook Question
Find the derivatives of the functions in Exercises 1–42.
𝓻 = ( sin θ )²
( cos θ - 1 )
Textbook Question
In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
y = 4 − x², (−1, 3)