Ch. 3 - Derivatives

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

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.3

Chapter 3, Problem 3.3

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

𝔂 = x³ - 3 (x² + π²)

Verified step by step guidance

Identify the function to differentiate: 𝔂 = x³ - 3(x² + π²).

Apply the power rule to differentiate x³. The power rule states that the derivative of x^n is n*x^(n-1).

Differentiate the term -3(x² + π²). Use the constant multiple rule, which allows you to take the constant outside the derivative, and then apply the power rule to x².

Recognize that π² is a constant, and the derivative of a constant is zero.

Combine the derivatives of each term to find the derivative of the entire function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.

Power Rule

The power rule is a fundamental technique for finding derivatives of polynomial functions. It states that if f(x) = x^n, where n is a real number, then the derivative f'(x) is given by n*x^(n-1). This rule simplifies the differentiation process, allowing for quick calculations of derivatives for terms involving powers of x.

Constant Rule

The constant rule in differentiation states that the derivative of a constant is zero. This means that if a function includes a constant term, it does not affect the slope of the function at any point. For example, in the function y = x³ - 3(x² + π²), the term -3π² is a constant, and its derivative contributes nothing to the overall derivative of the function.

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Related Practice

Textbook Question

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

g(x) = { x²/³, x ≥ 0

x¹/³, x < 0

219

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Textbook Question

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

______

𝓻 = √2θ sinθ

278

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Textbook Question

a. Graph the function

ƒ(x) = { x², -1 ≤ x < 0

{ -x², 0 ≤ x ≤ 1.

b. Is ƒ continuous at x = 0?

c. Is ƒ differentiable at x = 0?

Give reasons for your answers.

155

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Textbook Question

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

𝔂 = 1 x² csc 2

2 x

281

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Textbook Question

Slopes, Tangent Lines, and Normal Lines

In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.

x²y² = 9, (–1,3)

232

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Textbook Question

Is there a value of b that will make

g(x) = { x + b, x < 0

cos x, x ≥ 0

continuous at x = 0? Differentiable at x = 0? Give reasons for your answers.

227

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