Skip to main content
Ch. 4 - Applications of the Derivative
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. 4 - Applications of the DerivativeProblema 36
Capitolo 4, Problema 36

Evaluate and simplify y'.
y = (3t²−1 / 3t²+1)^−3

Guida verificata passo dopo passo
1
Step 1: Recognize that the function y = \(\left\)(\(\frac{3t^2 - 1}{3t^2 + 1}\)\(\right\))^{-3} is a composition of functions, specifically a power function and a rational function. To find y', we will use the chain rule.
Step 2: Apply the chain rule. The chain rule states that if y = f(g(t)), then y' = f'(g(t)) \(\cdot\) g'(t). Here, let u = \(\frac{3t^2 - 1}{3t^2 + 1}\), so y = u^{-3}.
Step 3: Differentiate the outer function with respect to u. The derivative of u^{-3} with respect to u is -3u^{-4}.
Step 4: Differentiate the inner function u = \(\frac{3t^2 - 1}{3t^2 + 1}\) with respect to t using the quotient rule. The quotient rule states that if u = \(\frac{v}{w}\), then u' = \(\frac{v'w - vw'}{w^2}\). Here, v = 3t^2 - 1 and w = 3t^2 + 1.
Step 5: Combine the results from Steps 3 and 4. Substitute the derivative of the inner function into the chain rule expression from Step 2 to find y'. Simplify the expression if possible.

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:
6m

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, representing the slope of the tangent line to the curve at any given point. In this context, we need to apply the rules of differentiation to find y', the derivative of the given function y.
Video consigliato:
Percorso guidato
05:44
Derivatives

Chain Rule

The chain rule is a formula for computing the derivative of a composite function. It states that if a function y can be expressed as a composition of two functions, say u(t) and v(u), then the derivative of y with respect to t is the product of the derivative of y with respect to u and the derivative of u with respect to t. This rule is essential for differentiating functions raised to a power, as seen in the given expression.
Video consigliato:
05:02
Intro to the Chain Rule

Quotient Rule

The quotient rule is used to differentiate functions that are expressed as the ratio of two other functions. It states that if y = f(t)/g(t), then the derivative y' is given by (f'g - fg')/g². In this problem, since y is a fraction, applying the quotient rule will be necessary to correctly differentiate the numerator and denominator before simplifying the result.
Video consigliato:
06:43
The Quotient Rule
Pratica correlata
Domanda del libro di testo

Use the guidelines of this section to make a complete graph of f.

f(x) = 2 - 2x2/3 + x4/3

218
views
Domanda del libro di testo

Use the guidelines of this section to make a complete graph of f.

f(x) = tan⁻¹ (x²/√3)

237
views
Domanda del libro di testo

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

√146

278
views
Domanda del libro di testo

Use the guidelines of this section to make a complete graph of f.

f(x) = x + 2 cos x on [-2π,2π)

283
views
Domanda del libro di testo

Demand and elasticity The economic advisor of a large tire store proposes the demand function D(p) = 1800/p-40, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p.

c. Find the elasticity function on the domain of the demand function.

294
views
Domanda del libro di testo

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = (x2 + 12)/(2x + 1)

269
views