Ch. 7 - Transcendental Functions

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

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.3.17

Capitolo 7, Problema 7.3.17

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = cos(e^(-θ^2))

Guida verificata passo dopo passo

Identify the outer function and the inner function in the composition. Here, the outer function is \(\cos(u)\) where \(u = e^{-\theta^2}\).

Recall the chain rule for derivatives: if \(y = f(g(\theta))\), then \(\frac{dy}{d\theta} = f'(g(\theta)) \cdot g'(\theta)\).

Differentiate the outer function with respect to its argument \(u\): \(\frac{d}{du} \cos(u) = -\sin(u)\).

Differentiate the inner function \(u = e^{-\theta^2}\) with respect to \(\theta\). Use the chain rule again: \(\frac{d}{d\theta} e^{-\theta^2} = e^{-\theta^2} \cdot \frac{d}{d\theta}(-\theta^2) = e^{-\theta^2} \cdot (-2\theta)\).

Combine the results using the chain rule: \(\frac{dy}{d\theta} = -\sin(e^{-\theta^2}) \cdot e^{-\theta^2} \cdot (-2\theta)\). Simplify the expression if needed.

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Chain Rule

The chain rule is a fundamental differentiation technique used when a function is composed of other 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. For example, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

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Intro to the Chain Rule

Derivative of the Cosine Function

The derivative of the cosine function with respect to its variable is the negative sine of that variable. Specifically, if y = cos(u), then dy/du = -sin(u). This rule is essential when differentiating trigonometric functions, especially when combined with the chain rule.

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Derivatives of Sine & Cosine

Derivative of the Exponential Function

The derivative of the exponential function e^u with respect to its variable is e^u times the derivative of u. That is, if y = e^u, then dy/dx = e^u * du/dx. This property is crucial when differentiating expressions where the exponent itself is a function of the variable.

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Derivatives of General Exponential Functions

Pratica correlata

Domanda del libro di testo

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

27. y = (1 - θ)tanh⁻¹(θ)

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Domanda del libro di testo

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

17. lim(x→∞)arcsec(x)

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17. Show that √(10x+1) and √(x+1) grow at the same rate as x→∞ by showing that they both grow at the same rate as √x as x→∞.

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Domanda del libro di testo

Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.

sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞

cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1

tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1

sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1

csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1

coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1

Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.

63. tanh⁻¹(-1/2)

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Domanda del libro di testo

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

31. y = cos⁻¹(x) - x sech⁻¹(x)