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 3.8.19a
Capitolo 3, Problema 3.8.19a

13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
cos y = x; (0, π/2)

Guida verificata passo dopo passo
1
Start by differentiating both sides of the equation with respect to x. The equation is cos(y) = x.
Differentiate the left side: The derivative of cos(y) with respect to x is -sin(y) * (dy/dx) because of the chain rule.
Differentiate the right side: The derivative of x with respect to x is 1.
Set the derivatives equal to each other: -sin(y) * (dy/dx) = 1.
Solve for dy/dx by isolating it: dy/dx = -1/sin(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:
1m

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. Instead of solving for y in terms of x, we differentiate both sides of the equation with respect to x, treating y as a function of x. This method is particularly useful for equations that are difficult or impossible to rearrange.
Video consigliato:
Percorso guidato
05:14
Finding The Implicit Derivative

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When using implicit differentiation, the chain rule is applied to account for the derivative of y with respect to x, denoted as dy/dx. This means that when differentiating a function of y, we multiply by dy/dx to reflect the dependence of y on x.
Video consigliato:
05:02
Intro to the Chain Rule

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are essential in calculus for modeling periodic phenomena and solving various problems. In the context of the given equation, cos(y) = x, understanding the properties and derivatives of these functions is crucial for finding dy/dx. The derivative of cos(y) involves the chain rule and is equal to -sin(y) * dy/dx.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions
Pratica correlata
Domanda del libro di testo

Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>

a. Estimate L' (1.5) and state the physical meaning of this quantity.

190
views
Domanda del libro di testo

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

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

223
views
Domanda del libro di testo

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = √2x+1; a= 4

182
views
Domanda del libro di testo

Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>


a. Express the angle of elevation θ from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80−h.)

266
views
Domanda del libro di testo

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(t) = 3t⁴; a= -2, 2

257
views
Domanda del libro di testo

Use the graph of f in the figure to do the following. <IMAGE>

a. Find the values of x in (-2,2) at which f is not continuous.

321
views