Skip to main content
Ch. 7 - Transcendental Functions
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. 7 - Transcendental FunctionsProblema 7.3.124
Capitolo 7, Problema 7.3.124

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
124. x^(sin y) = ln y

Guida verificata passo dopo passo
1
Start with the given equation: \(x^{\sin y} = \ln y\).
Since the equation involves both \(x\) and \(y\) in a complicated way, use implicit differentiation with respect to \(x\). Remember that \(y\) is a function of \(x\), so when differentiating terms involving \(y\), apply the chain rule.
Differentiate the left side \(x^{\sin y}\) using logarithmic differentiation: first take the natural logarithm of both sides to simplify the exponentiation. That is, write \(\ln(x^{\sin y}) = \ln(\ln y)\).
Simplify the left side using logarithm properties: \(\sin y \cdot \ln x = \ln(\ln y)\).
Now differentiate both sides with respect to \(x\). For the left side, use the product rule on \(\sin y \cdot \ln x\), remembering that \(y\) depends on \(x\). For the right side, use the chain rule on \(\ln(\ln y)\).

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:
5m

Concetti chiave

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

Implicit Differentiation

Implicit differentiation is used when a function is defined implicitly rather than explicitly. It involves differentiating both sides of an equation with respect to the independent variable, treating dependent variables as functions of that variable, and then solving for the derivative.
Video consigliato:
Percorso guidato
05:14
Finding The Implicit Derivative

Logarithmic Differentiation

Logarithmic differentiation simplifies differentiation of functions where variables appear as exponents or products by taking the natural logarithm of both sides. This transforms complicated expressions into sums or products, making differentiation more manageable.
Video consigliato:
06:30
Logarithmic Differentiation

Chain Rule

The chain rule is a fundamental differentiation technique used when differentiating composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Video consigliato:
05:02
Intro to the Chain Rule
Pratica correlata
Domanda del libro di testo

37. Plutonium-239 The half-life of the plutonium isotope is 24,360 years. If 10 g of plutonium is released into the atmosphere by a nuclear accident, how many years will it take for 80% of the isotope to decay?

4
views
Domanda del libro di testo

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

63. lim (x → ∞) ((x + 2)/(x - 1))^x

18
views
Domanda del libro di testo

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = e^(cost+lnt)

21
views
Domanda del libro di testo

132. What is special about the functions

f(x) = arcsin((1/√(x²+1)) and g(x)=arctan(1/x)?

Explain.

26
views
Domanda del libro di testo

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

35. lim (x → 0⁺) ln(x² + 2x) / ln x

52
views
Domanda del libro di testo

135. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e^(2x), below by the curve y = e^x, and on the right by the line x = ln(3).

23
views