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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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