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

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Identify the function to differentiate: \(y = e^{\theta}(\sin\theta + \cos\theta)\), which is a product of two functions of \(\theta\).

Recall the product rule for differentiation: if \(y = u(\theta) \cdot v(\theta)\), then \(\frac{dy}{d\theta} = u'(\theta) v(\theta) + u(\theta) v'(\theta)\).

Set \(u(\theta) = e^{\theta}\) and \(v(\theta) = \sin\theta + \cos\theta\). Compute their derivatives separately: \(u'(\theta) = e^{\theta}\) and \(v'(\theta) = \cos\theta - \sin\theta\).

Apply the product rule: \(\frac{dy}{d\theta} = e^{\theta}(\sin\theta + \cos\theta) + e^{\theta}(\cos\theta - \sin\theta)\).

Simplify the expression by combining like terms inside the parentheses to write the derivative in its simplest form.

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Key Concepts

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

Derivative of Exponential Functions

The derivative of an exponential function like e^θ with respect to θ is e^θ itself. This property is fundamental when differentiating expressions where the exponential function is multiplied by other functions.

Product Rule

The product rule is used to differentiate the product of two functions. It states that the derivative of f(θ)g(θ) is f'(θ)g(θ) + f(θ)g'(θ). This rule is essential when differentiating y = e^θ(sinθ + cosθ).

Derivatives of Trigonometric Functions

The derivatives of sinθ and cosθ with respect to θ are cosθ and -sinθ, respectively. Knowing these derivatives is crucial for differentiating the trigonometric part of the function y = e^θ(sinθ + cosθ).

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