Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 3 - Derivatives

Problema 64

Capitolo 3, Problema 64

Calculate the derivative of the following functions.
y = (e^x / x+1)⁸

Guida verificata passo dopo passo

Step 1: Recognize that the function y = \(\left\)(\(\frac{e^x}{x+1}\)\(\right\))^8 is a composite function, which means we will need to use the chain rule to find its derivative.

Step 2: Let u = \(\frac{e^x}{x+1}\). Then, y = u^8. The chain rule states that \(\frac{dy}{dx}\) = \(\frac{dy}{du}\) \(\cdot\) \(\frac{du}{dx}\). First, find \(\frac{dy}{du}\) = 8u^7.

Step 3: Now, find \(\frac{du}{dx}\). Since u = \(\frac{e^x}{x+1}\), use the quotient rule: \(\frac{du}{dx}\) = \(\frac{(x+1)\cdot e^x - e^x \cdot 1}{(x+1)^2}\).

Step 4: Simplify \(\frac{du}{dx}\) to get \(\frac{e^x(x+1) - e^x}{(x+1)^2}\) = \(\frac{e^x(x+1-1)}{(x+1)^2}\) = \(\frac{e^x x}{(x+1)^2}\).

Step 5: Substitute \(\frac{du}{dx}\) and \(\frac{dy}{du}\) back into the chain rule formula: \(\frac{dy}{dx}\) = 8u^7 \(\cdot\) \(\frac{e^x x}{(x+1)^2}\). Finally, replace u with \(\frac{e^x}{x+1}\) to express the derivative in terms of 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:

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 a fundamental concept in calculus that provides the slope of the tangent line to the curve of the function at any given point. The derivative is often denoted as f'(x) or dy/dx and can be calculated using various rules, such as the power rule, product rule, and quotient rule.

Video consigliato:

Percorso guidato

05:44

Derivatives

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when dealing with functions raised to a power, as in the given problem.

Video consigliato:

05:02

Intro to the Chain Rule

Quotient Rule

The quotient rule is used to find 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 (v * du/dx - u * dv/dx) / v². This rule is essential for differentiating functions like y = (e^x)/(x+1), where the numerator and denominator are both functions that need to be differentiated separately.

Video consigliato:

06:43

The Quotient Rule

Pratica correlata

Domanda del libro di testo

Calculate the derivative of the following functions.

y = tan(xe^x)

416

views

Domanda del libro di testo

A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.

y=3x−4; P(1, −1)

406

views

Domanda del libro di testo

Population growth Consider the following population functions.

a. Find the instantaneous growth rate of the population, for t≥0.

p(t) = 600 (t²+3/t²+9)

298

views

Domanda del libro di testo

y = 2/x; P(1, 2)

296