Domanda del libro di testo
Derivatives
In Exercises 1–18, find dy/dx.
f(x) = x³ sin x cos x
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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 3, Problema 3.5.31
Derivatives
In Exercises 27–32, find dp/dq.
p = (q sin q) / (q² − 1)
Guida verificata passo dopo passo1
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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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.
Video consigliato:
Percorso guidato
05:44
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.
Video consigliato:
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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Percorso guidato
6:04Introduction to Trigonometric Functions
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