Skip to main content
Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 3.5.57
Capitolo 3, Problema 3.5.57

Find y'' for the following functions.
y = x sin x

Guida verificata passo dopo passo
1
First, identify that the function y = x sin(x) is a product of two functions: u(x) = x and v(x) = sin(x). To find the first derivative y', use the product rule: (uv)' = u'v + uv'.
Calculate the derivative of u(x) = x, which is u'(x) = 1.
Calculate the derivative of v(x) = sin(x), which is v'(x) = cos(x).
Apply the product rule: y' = u'v + uv' = (1)(sin(x)) + (x)(cos(x)) = sin(x) + x cos(x).
To find the second derivative y'', differentiate y' = sin(x) + x cos(x) with respect to x. Use the sum rule and the product rule again for the term x cos(x). The derivative of sin(x) is cos(x), and for x cos(x), apply the product rule: (x cos(x))' = (1)(cos(x)) + (x)(-sin(x)) = cos(x) - x sin(x). Therefore, y'' = cos(x) + (cos(x) - x sin(x)).

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:
7m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, we need to differentiate the function y = x sin x to find its first derivative, y', and then differentiate y' to find the second derivative, y''. This process is fundamental in calculus for analyzing the behavior of functions.
Video consigliato:
Percorso guidato
05:53
Finding Differentials

Product Rule

The Product Rule is a formula used to differentiate products of two functions. It states that if you have two functions u(x) and v(x), the derivative of their product is given by u'v + uv'. In the case of y = x sin x, we will apply the Product Rule to differentiate the function correctly, ensuring we account for both components of the product.
Video consigliato:
05:18
The Product Rule

Second Derivative

The second derivative of a function, denoted as y'', is the derivative of the first derivative, y'. It provides information about the curvature of the function and can indicate points of inflection. In this problem, finding y'' will help us understand how the rate of change of y = x sin x evolves, which is crucial for analyzing its behavior in applications such as motion or optimization.
Video consigliato:
06:02
The Second Derivative Test: Finding Local Extrema
Pratica correlata
Domanda del libro di testo

Piston compression A piston is seated at the top of a cylindrical chamber with radius 5 cm when it starts moving into the chamber at a constant speed of 3 cm/s (see figure). What is the rate of change of the volume of the cylinder when the piston is 2 cm from the base of the chamber? <IMAGE>

291
views
Domanda del libro di testo

67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.


f(x) = 3x-4

191
views
Domanda del libro di testo

27–76. Calculate the derivative of the following functions.

y=10x+1y=\(\sqrt{10x+1}\)

423
views
Domanda del libro di testo

Find the following higher-order derivatives.


d²/dx² (In(x² + 1))

190
views
Domanda del libro di testo

Find (f^−1)′(3), where f(x)=x³+x+1.

203
views
Domanda del libro di testo

Suppose f is a one-to-one function with f(2)=8 and f′(2)=4. What is the value of (f^−1)′(8)?

192
views