Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 3 - Derivatives

Problema 3.54

Capitolo 3, Problema 3.54

In Exercises 53 and 54, find dr/ds.

2rs - r - s + s² = -3

Guida verificata passo dopo passo

First, identify the equation given: 2rs - r - s + s² = -3. We need to find the derivative dr/ds.

To find dr/ds, we will use implicit differentiation. Differentiate both sides of the equation with respect to s.

When differentiating 2rs with respect to s, apply the product rule: d/ds(2rs) = 2r(ds/ds) + 2s(dr/ds).

Differentiate the remaining terms: d/ds(-r) = -dr/ds, d/ds(-s) = -1, and d/ds(s²) = 2s.

Set the derivative of the right side of the equation to zero, as d/ds(-3) = 0. Combine all differentiated terms to form an equation involving dr/ds, and solve for dr/ds.

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.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In this case, we differentiate both sides of the equation with respect to one variable while treating the other variable as a function of the first. This allows us to find derivatives like dr/ds even when r and s are intertwined.

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 applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. In the context of finding dr/ds, it helps us relate the rates of change of r with respect to s, especially when r is expressed as a function of s.

Video consigliato:

05:02

Intro to the Chain Rule

Solving for Derivatives

Solving for derivatives involves isolating the desired derivative after applying differentiation techniques. In this problem, after differentiating the given equation implicitly, we will rearrange the resulting equation to express dr/ds explicitly. This step is crucial for obtaining the final answer and understanding the relationship between the variables.

Video consigliato:

5:02

Solving Logarithmic Equations

Pratica correlata

Domanda del libro di testo

The diameter of a sphere is measured as 100 ± 1 cm and the volume is calculated from this measurement. Estimate the percentage error in the volume calculation.

240

views

Domanda del libro di testo

Second Derivatives

In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.

x²/³ + y²/³ = 1

220

views

Domanda del libro di testo

Find the first and second derivatives of the functions in Exercises 33–38.

w = ((1 + 3z) / 3z) (3 − z)

196