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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 3.3.65
Capitolo 3, Problema 3.3.65

Cylinder pressure If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form
P = (nRT / (V − nb)) − (an² / V²),
in which a, b, n, and R are constants. Find dP/dV. (See accompanying figure.)


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Guida verificata passo dopo passo
1
Identify the given formula for pressure P in terms of volume V: P = (nRT / (V − nb)) − (an² / V²).
Recognize that you need to find the derivative of P with respect to V, denoted as dP/dV.
Apply the derivative rules: For the first term (nRT / (V − nb)), use the quotient rule, which states that if you have a function f(x) = u(x)/v(x), then its derivative f'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))².
For the second term (an² / V²), apply the power rule for derivatives, which states that if you have a function f(x) = x^n, then its derivative f'(x) = n*x^(n-1).
Combine the derivatives of both terms to express dP/dV as a single expression, ensuring to simplify where possible.

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Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function changes with respect to one of its variables. In this context, dP/dV represents the derivative of pressure P with respect to volume V, indicating how pressure changes as volume changes. Understanding differentiation is crucial for solving problems involving rates of change.
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Percorso guidato
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Finding Differentials

Product Rule

The product rule is a technique used in calculus to differentiate functions that are products of two or more functions. It states that the derivative of a product of two functions u and v is given by (du/dx)v + u(dv/dx). In the given formula, the pressure P is expressed as a product of terms involving V, requiring the application of the product rule to find dP/dV.
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The Product Rule

Chain Rule

The chain rule is a method for differentiating composite functions, where one function is nested within another. It is essential when dealing with expressions where variables are interdependent. In the formula for P, terms like (V - nb) and V² involve nested functions, necessitating the use of the chain rule to correctly compute the derivative dP/dV.
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Intro to the Chain Rule
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