Domanda del libro di testo
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
views
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 3, Problema 3.3
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x³ - 3 (x² + π²)
Guida verificata passo dopo passo1
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.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
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.
Video consigliato:
Percorso guidato
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.
Video consigliato:
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.
Video consigliato:
04:02The Power Rule
Pratica correlata
Domanda del libro di testo
Find the derivatives of the functions in Exercises 1–42.
______
𝓻 = √2θ sinθ
278
views
Domanda del libro di testo
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.
156
views
Domanda del libro di testo
Find the derivatives of the functions in Exercises 1–42.
𝔂 = 1 x² csc 2
2 x
281
views
Domanda del libro di testo
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
views
Domanda del libro di testo
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
views