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

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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).

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.

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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Related Practice

Textbook Question

Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.

f(x) = (x + b) / (x − 2), b > −2 and constant

Textbook Question

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76.

Try it—you just keep on cycling. Find the limits some other way.

69. lim (x → ∞) (√(9x + 1)) / (√(x + 1))

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Textbook Question

Let f(x) = x³ − 3x² − 1, x ≥ 2. Find the value of df⁻¹/dx at the point x = −1 = f(3).

Textbook Question

Evaluate the integrals in Exercises 33–54.

53. ∫ (e^r / (1 + e^r)) dr