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
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.3
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x³ - 3 (x² + π²)
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
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.
Recommended video:
05:44
Derivatives
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.
Recommended video:
Guided course
5:50Power Rules
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.
Recommended video:
04:02The Power Rule
Related Practice
Textbook Question
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x cos(2x + 3y) = y sin x
Textbook Question
Show that the linearization of f(x) = (1 + x)ᵏ at x = 0 is L(x) = 1 + kx.
Textbook Question
Find the derivatives of all orders of the functions in Exercises 29–32.
y = x⁵ / 120
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
𝔂 = 1 x² csc 2
2 x
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)