Domanda del libro di testo
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
12. lim (x → ∞) (x - 8x²) / (12x² + 5x)
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.2.64
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
64. y = 1/(t(t+1)(t+2))
Guida verificata passo dopo passo1
Start by expressing the function clearly: \(y = \frac{1}{t(t+1)(t+2)}\). This can also be written as \(y = [t(t+1)(t+2)]^{-1}\) to prepare for logarithmic differentiation.
Take the natural logarithm of both sides to simplify the product inside the logarithm: \(\ln y = \ln \left( [t(t+1)(t+2)]^{-1} \right)\).
Use the logarithm power rule to bring down the exponent: \(\ln y = -1 \cdot \ln [t(t+1)(t+2)]\).
Apply the logarithm product rule to expand the right side: \(\ln y = - (\ln t + \ln (t+1) + \ln (t+2))\).
Differentiate both sides with respect to \(t\). Remember that \(\frac{d}{dt} (\ln y) = \frac{1}{y} \frac{dy}{dt}\) by the chain rule, and differentiate each logarithmic term on the right side accordingly.
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Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers of variables by taking the natural logarithm of both sides. This simplifies the differentiation process by converting multiplication into addition and division into subtraction, making it easier to handle complex expressions.
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06:30
Logarithmic Differentiation
Properties of Logarithms
The properties of logarithms, such as log(ab) = log a + log b and log(a/b) = log a - log b, allow us to break down complicated products and quotients into sums and differences. These properties are essential in logarithmic differentiation to simplify the function before differentiating.
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05:36Change of Base Property
Implicit Differentiation
Implicit differentiation involves differentiating both sides of an equation with respect to the independent variable, treating the dependent variable as a function of that variable. In logarithmic differentiation, after taking logs, implicit differentiation is used to find dy/dx by differentiating the transformed equation.
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Percorso guidato
05:14Finding The Implicit Derivative
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