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 3.8.52
Capitolo 3, Problema 3.8.52

51–56. Second derivatives Find d²y/dx².
2x²+y² = 4

Guida verificata passo dopo passo
1
First, identify the given equation: \(2x^2 + y^2 = 4\). This is an implicit function of \(x\) and \(y\).
Differentiate both sides of the equation with respect to \(x\) to find the first derivative \(\frac{dy}{dx}\). Use implicit differentiation: \(\frac{d}{dx}(2x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(4)\).
Calculate the derivatives: \(\frac{d}{dx}(2x^2) = 4x\) and \(\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}\). The right side derivative is zero since \(4\) is a constant.
Set up the equation from the derivatives: \(4x + 2y \frac{dy}{dx} = 0\). Solve for \(\frac{dy}{dx}\) to find the first derivative.
Differentiate \(\frac{dy}{dx}\) again with respect to \(x\) to find \(\frac{d^2y}{dx^2}\). Use the quotient rule or implicit differentiation as needed, and substitute \(\frac{dy}{dx}\) from the previous step into this new derivative.

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:
9m

Concetti chiave

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

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In this case, the equation 2x² + y² = 4 involves both x and y, making it necessary to differentiate both sides with respect to x while treating y as a function of x. This allows us to find the first derivative dy/dx before proceeding to the second derivative.
Video consigliato:
Percorso guidato
05:14
Finding The Implicit Derivative

First Derivative

The first derivative, denoted as dy/dx, represents the rate of change of the dependent variable y with respect to the independent variable x. It provides information about the slope of the tangent line to the curve defined by the equation at any given point. To find the second derivative, we first need to compute the first derivative using implicit differentiation.
Video consigliato:
07:09
The First Derivative Test: Finding Local Extrema

Second Derivative

The second derivative, denoted as d²y/dx², measures the rate of change of the first derivative. It provides insights into the curvature of the function, indicating whether the function is concave up or concave down at a given point. To find d²y/dx², we differentiate the first derivative again, applying implicit differentiation as necessary to account for the relationship between x and y.
Video consigliato:
06:02
The Second Derivative Test: Finding Local Extrema
Pratica correlata
Domanda del libro di testo

Find d/dx(ln(x/x²+1)) without using the Quotient Rule.

193
views
Domanda del libro di testo

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 7x) / 3x

311
views
Domanda del libro di testo

Find f′(x), f′′(x), and f′′′(x) for the following functions.

f(x) = 3x2 + 5ex

255
views
Domanda del libro di testo

Derivatives Find and simplify the derivative of the following functions.

f(x) = 2e^x-1 / 2e^x+1

331
views
Domanda del libro di testo

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

Suppose the speed of a car approaching a stop sign is given by v (t) = (t-5)², for 0 ≤ t ≤ 5, where t is measured in seconds and v(t) is measured in meters per second. Find v′(3).

214
views
Domanda del libro di testo

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

g(x) = 6x⁵ - 5/2 x² + x + 5

286
views