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 3.8.46b
Capitolo 3, Problema 3.8.46b

45–50. Tangent lines Carry out the following steps. <IMAGE>
b. Determine an equation of the line tangent to the curve at the given point.
x³+y³=2xy; (1, 1)

Guida verificata passo dopo passo
1
First, identify the given curve equation: \(x^3 + y^3 = 2xy\). We need to find the derivative to determine the slope of the tangent line at the point (1, 1).
Use implicit differentiation to differentiate both sides of the equation with respect to \(x\). Remember that \(y\) is a function of \(x\), so apply the chain rule when differentiating terms involving \(y\).
Differentiate the left side: \(\frac{d}{dx}(x^3) + \frac{d}{dx}(y^3) = 3x^2 + 3y^2 \frac{dy}{dx}\).
Differentiate the right side: \(\frac{d}{dx}(2xy) = 2y + 2x \frac{dy}{dx}\).
Set the derivatives equal: \(3x^2 + 3y^2 \frac{dy}{dx} = 2y + 2x \frac{dy}{dx}\). Solve for \(\frac{dy}{dx}\) to find the slope of the tangent line at the point (1, 1). Substitute \(x = 1\) and \(y = 1\) into the derivative to find the specific slope at that point. Finally, use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), to write the equation of the tangent line.

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.

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation, rather than explicitly as y = f(x). In this case, the equation x³ + y³ = 2xy involves both x and y, requiring us to differentiate both sides with respect to x while treating y as a function of x. This method allows us to find dy/dx, which is essential for determining the slope of the tangent line.
Video consigliato:
Percorso guidato
05:14
Finding The Implicit Derivative

Tangent Line Equation

The equation of a tangent line at a given point on a curve can be expressed using the point-slope form: y - y₀ = m(x - x₀), where (x₀, y₀) is the point of tangency and m is the slope at that point. Once the derivative (slope) is calculated using implicit differentiation, this formula can be applied to find the specific equation of the tangent line at the point (1, 1) for the given curve.
Video consigliato:
Percorso guidato
05:14
Equations of Tangent Lines

Slope of the Tangent Line

The slope of the tangent line represents the instantaneous rate of change of the function at a specific point. In the context of the curve defined by the equation x³ + y³ = 2xy, the slope can be found by evaluating the derivative dy/dx at the point (1, 1). This slope is crucial for constructing the tangent line, as it indicates how steep the line will be at that point on the curve.
Video consigliato:
Percorso guidato
05:13
Slopes of Tangent Lines
Pratica correlata
Domanda del libro di testo

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.


b. ln(x + 1) + ln(x − 1) = ln(x² − 1), for all x.

155
views
Domanda del libro di testo

The Chain Rule for second derivatives

b. Use the formula in part (a) to calculate d2dx2(sin(3x4+5x2+2))\(\frac{d^2}{dx^2}\[\left\)(\(\sin\]\left\)(3x^4+5x^2+2\(\right\))\(\right\)).

408
views
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counter example.

b. d²/dx² (sin x) = sin x.

176
views
Domanda del libro di testo

{Use of Tech} Spring oscillations A spring hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance x (in inches) of the mass from its equilibrium position after t seconds is given by the function x(t) = 10sin t - 10cos t, where x is positive when the mass is above the equilibrium position. <IMAGE>

b. Find dx/dt and interpret the meaning of this derivative.  

208
views
Domanda del libro di testo

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(x) = 1/x+1; a = -1/2;5

317
views
Domanda del libro di testo

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

limx→04+sin(x)−2x{\(\displaystyle\)\(\lim\)_{x\(\to\)0}}\(\frac{\sqrt{4+\sin\left(x\right)}\)-2}{x}

263
views