Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.3.12

Chapter 3, Problem 3.3.12

Derivative Calculations

In Exercises 1–12, find the first and second derivatives.

r = 12/θ − 4/θ³ + 1/θ⁴

Verified step by step guidance

Step 1: Identify the function for which you need to find the derivatives. The function given is \( r(\theta) = \frac{12}{\theta} - \frac{4}{\theta^3} + \frac{1}{\theta^4} \).

Step 2: Rewrite the function in terms of powers of \( \theta \) to make differentiation easier. This gives \( r(\theta) = 12\theta^{-1} - 4\theta^{-3} + \theta^{-4} \).

Step 3: Find the first derivative \( r'(\theta) \) by applying the power rule \( \frac{d}{d\theta}[\theta^n] = n\theta^{n-1} \) to each term. Differentiate each term separately.

Step 4: Find the second derivative \( r''(\theta) \) by differentiating \( r'(\theta) \) using the power rule again. Apply the rule to each term of the first derivative.

Step 5: Simplify the expressions for both the first and second derivatives to obtain the final forms. Ensure all terms are expressed in terms of \( \theta \) with negative exponents as needed.

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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 determine how a function behaves locally. The first derivative indicates the slope of the tangent line to the curve at any point, while the second derivative provides information about the curvature or concavity of the function.

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 x (i.e., y = f(u) and u = g(x)), then the derivative of y with respect to x is the product of the derivative of f with respect to u and the derivative of g with respect to x. This rule is essential when differentiating functions that involve variables in the denominator, as seen in the given expression.

Power Rule

The power rule is a basic technique for finding the derivative of a function in the form of x^n, where n is any real number. According to this rule, the derivative of x^n is n*x^(n-1). This rule simplifies the process of differentiation, especially for polynomial and rational functions, making it crucial for calculating the first and second derivatives of the given function.

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Find the slope of the curve x³y³ + y² = x + y at the points (1, 1) and (1, -1).

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Heating a plate When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm/min. At what rate is the plate’s area increasing when the radius is 50 cm?

Textbook Question

In Exercises 41–58, find dy/dt.

y = sin²(πt − 2)

Textbook Question

Using the Alternative Formula for Derivatives

Use the formula

f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)

to find the derivative of the functions in Exercises 23–26.

g(x) = x / (x − 1)

Textbook Question

Find the derivatives of the functions in Exercises 1–42.

s = √ t .