In Exercises 19–22, find the values of the derivatives.

dr/dθ |θ₌₀ if r = 2/√(4 – θ)

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First, identify the function given: \( r(\theta) = \frac{2}{\sqrt{4 - \theta}} \). We need to find the derivative \( \frac{dr}{d\theta} \) and evaluate it at \( \theta = 0 \).

To differentiate \( r(\theta) \), recognize it as a composition of functions. Use the chain rule: if \( r(\theta) = f(g(\theta)) \), then \( \frac{dr}{d\theta} = f'(g(\theta)) \cdot g'(\theta) \).

Rewrite \( r(\theta) = (4 - \theta)^{-1/2} \) to make differentiation easier. Now, apply the chain rule: let \( u = 4 - \theta \), so \( r = u^{-1/2} \).

Differentiate \( r = u^{-1/2} \) with respect to \( u \): \( \frac{dr}{du} = -\frac{1}{2}u^{-3/2} \). Then, differentiate \( u = 4 - \theta \) with respect to \( \theta \): \( \frac{du}{d\theta} = -1 \).

Combine the derivatives using the chain rule: \( \frac{dr}{d\theta} = \frac{dr}{du} \cdot \frac{du}{d\theta} = -\frac{1}{2}(4 - \theta)^{-3/2} \cdot (-1) \). Evaluate this expression at \( \theta = 0 \) to find the derivative at that point.

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주요 개념

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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 determine how a function behaves as its input changes. The notation dr/dθ indicates the derivative of the function r with respect to the variable θ, which is essential for understanding how r varies as θ changes.

Chain Rule

The chain rule is a formula for computing the derivative of a composite function. It states that if a function y is composed of two functions u and v, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is particularly useful when differentiating functions that are expressed in terms of other functions, such as r(θ).

Evaluating Derivatives at a Point

Evaluating a derivative at a specific point involves substituting the value of the variable into the derivative function. In this case, we need to find dr/dθ at θ = 0. This process provides the instantaneous rate of change of the function r at that particular value of θ, which is crucial for understanding the behavior of the function at that point.

In Exercises 19–22, find the values of the derivatives.dr/dθ |θ₌₀ if r = 2/√(4 – θ)