Domanda del libro di testo
Derivative Calculations
In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).
y = cos u, u = −x/3
263
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 3.6.63
Second Derivatives
Find y'' in Exercises 59–64.
y = x(2x + 1)⁴
Guida verificata passo dopo passo1
First, identify the function y = x(2x + 1)^4. We need to find the second derivative y''.
To find the first derivative y', apply the product rule. The product rule states that if you have a function y = u(x)v(x), then y' = u'(x)v(x) + u(x)v'(x). Here, let u(x) = x and v(x) = (2x + 1)^4.
Calculate the derivatives: u'(x) = 1 and v'(x) using the chain rule. For v(x) = (2x + 1)^4, let w = 2x + 1, then v(x) = w^4. The derivative v'(x) = 4w^3 * (dw/dx) = 4(2x + 1)^3 * 2.
Substitute the derivatives back into the product rule: y' = 1 * (2x + 1)^4 + x * 8(2x + 1)^3.
Simplify y' and then find the second derivative y'' by differentiating y' again. Use the product rule and chain rule as necessary to find y''.
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:
9mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Product Rule
The product rule is a fundamental differentiation rule used when finding the derivative of a product of two functions. If you have two functions u(x) and v(x), the derivative of their product is given by u'(x)v(x) + u(x)v'(x). This rule is essential for differentiating expressions like y = x(2x + 1)⁴, where x is multiplied by another function.
Video consigliato:
05:18
The Product Rule
Chain Rule
The chain rule is used to differentiate composite functions, where one function is nested inside another. If you have a function y = f(g(x)), the derivative is f'(g(x)) * g'(x). In the given problem, the chain rule helps differentiate the term (2x + 1)⁴, treating it as a composite function with an inner function g(x) = 2x + 1.
Video consigliato:
05:02Intro to the Chain Rule
Second Derivative
The second derivative, denoted as y'', is the derivative of the first derivative y'. It provides information about the curvature or concavity of the function's graph. Calculating the second derivative involves differentiating the first derivative, often requiring the application of differentiation rules like the product and chain rules multiple times.
Video consigliato:
06:02The Second Derivative Test: Finding Local Extrema
Pratica correlata
Domanda del libro di testo
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x⁻¹/² sec (2x)²
239
views
Domanda del libro di testo
Find the derivatives of the functions in Exercises 19–40.
y = (4x + 3)⁴(x + 1)⁻³
221
views
Domanda del libro di testo
Are there any points on the curve y = x - 1/(2x) where the slope is 2? If so, find them.
212
views
Domanda del libro di testo
Power Rule for negative integers Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is,
d/dx (x⁻ᵐ) = −mx⁻ᵐ⁻¹
where m is a positive integer.
277
views
Domanda del libro di testo
Find the derivatives of the functions in Exercises 1–42.
𝔂 = (x + 1)² (x² + 2x)
203
views