Derivatives


In Exercises 27–32, find dp/dq.


p = (q sin q) / (q² − 1)

Verified step by step guidance

1. 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.
2. 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.
3. 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).
4. Step 4: Differentiate v = q^2 - 1 with respect to q. This is a simple power rule: \frac{d}{dq}(q^2 - 1) = 2q.
5. 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.

Key Concepts

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.

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.

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.