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 3.2.33
Capitolo 3, Problema 3.2.33

Evaluate dy/dx and dy/dx|x=2 if y= x+1/x+2

Guida verificata passo dopo passo
1
Step 1: Identify the function y = \(\frac{x+1}{x+2}\). This is a rational function, which is a ratio of two polynomials.
Step 2: Use the quotient rule to find \(\frac{dy}{dx}\). The quotient rule states that if y = \(\frac{u}{v}\), then \(\frac{dy}{dx}\) = \(\frac{v \cdot \frac{du}{dx}\) - u \(\cdot\) \(\frac{dv}{dx}\)}{v^2}, where u = x+1 and v = x+2.
Step 3: Differentiate the numerator and the denominator separately. For u = x+1, \(\frac{du}{dx}\) = 1. For v = x+2, \(\frac{dv}{dx}\) = 1.
Step 4: Substitute the derivatives into the quotient rule formula: \(\frac{dy}{dx}\) = \(\frac{(x+2) \cdot 1 - (x+1) \cdot 1}{(x+2)^2}\).
Step 5: Simplify the expression for \(\frac{dy}{dx}\) and then evaluate it at x = 2 by substituting x = 2 into the simplified expression.

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:
4m

Concetti chiave

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

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as dy/dx, representing the rate of change of y with respect to x.
Video consigliato:
Percorso guidato
05:44
Derivatives

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If y = u/v, where u and v are functions of x, the derivative is given by dy/dx = (v(du/dx) - u(dv/dx)) / v². This rule is essential when differentiating functions that involve division.
Video consigliato:
06:43
The Quotient Rule

Evaluating Derivatives at a Point

Evaluating a derivative at a specific point involves substituting the x-value into the derivative function obtained from differentiation. For example, if you find dy/dx and need to evaluate it at x=2, you substitute 2 into the derivative expression to find the slope of the tangent line to the curve at that point.
Video consigliato:
04:50
Critical Points
Pratica correlata
Domanda del libro di testo

Explain the difference between the average rate of change and the instantaneous rate of change of a function f.

472
views
Domanda del libro di testo

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→a (mx+b)=ma+b, for any constants a, b, and m

319
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.

h(t) = t²/2 + 1

302
views
Domanda del libro di testo

The speed of sound (in m/s) in dry air is approximated the function v(T) = 331 + 0.6T, where T is the air temperature (in degrees Celsius). Evaluate v' (T) and interpret its meaning.

339
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.

An object dropped from rest falls d(t)=16t² feet in t seconds. Find d′(4).

263
views
Domanda del libro di testo

Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>

d/dx (xg(x)) | x=2

233
views