Derivatives

In Exercises 23–26, find dr/dθ.

r = θ sin θ + cos θ

Verified step by step guidance

Step 1: Identify the function r(θ) given in the problem, which is r = θ sin(θ) + cos(θ).

Step 2: Apply the derivative rules to find dr/dθ. The function r(θ) is composed of two parts: θ sin(θ) and cos(θ). Use the product rule for θ sin(θ) and the derivative of cos(θ).

Step 3: Recall the product rule for derivatives, which states that if you have a function u(θ) * v(θ), the derivative is u'(θ) * v(θ) + u(θ) * v'(θ). Apply this to θ sin(θ), where u(θ) = θ and v(θ) = sin(θ).

Step 4: Calculate the derivative of θ sin(θ) using the product rule: u'(θ) = 1 and v'(θ) = cos(θ). Therefore, the derivative is 1 * sin(θ) + θ * cos(θ).

Step 5: Calculate the derivative of cos(θ), which is -sin(θ). Combine the derivatives from steps 4 and 5 to find dr/dθ: dr/dθ = sin(θ) + θ cos(θ) - sin(θ). Simplify the expression to get the final derivative.

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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 of change of a function with respect to a variable. In calculus, it is a fundamental concept that allows us to understand how a function behaves as its input changes. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a given point.

Parametric Equations

In this context, the equation r = θ sin θ + cos θ is a parametric equation where r is expressed in terms of the parameter θ. Parametric equations allow us to define curves in a more flexible way, using one or more parameters to describe the coordinates of points on the curve. Understanding how to differentiate these equations is crucial for finding derivatives with respect to the parameter.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. When dealing with functions that depend on other functions, the chain rule allows us to find the derivative of the outer function while multiplying it by the derivative of the inner function. This is particularly important when differentiating parametric equations, as it helps in managing the relationships between the variables involved.

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