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

검증된 단계별 안내

Identify the function y = (x^2 - 2x + 2) e^{x} as a product of two functions: u = (x^2 - 2x + 2) and v = e^{x}.

Recall the product rule for derivatives: if y = u v, then \( \frac{dy}{dx} = u' v + u v' \).

Find the derivative of u with respect to x: \( u = x^2 - 2x + 2 \), so \( u' = 2x - 2 \).

Find the derivative of v with respect to x: \( v = e^{x} \), so \( v' = e^{x} \).

Apply the product rule: \( \frac{dy}{dx} = (2x - 2) e^{x} + (x^2 - 2x + 2) e^{x} \).

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주요 개념

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

Product Rule

The product rule is used to differentiate functions that are products of two or more functions. It states that the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x). This rule is essential when differentiating y = (x^2 - 2x + 2)e^x, where both parts depend on x.

Derivative of Polynomial Functions

Polynomial functions like x^2 - 2x + 2 are differentiated term-by-term using the power rule. The power rule states that d/dx[x^n] = nx^(n-1). Understanding how to differentiate each term correctly is crucial for applying the product rule effectively.

Derivative of Exponential Functions

The derivative of the exponential function e^x is e^x itself. This property simplifies differentiation when e^x is part of the function. Recognizing this helps in applying the product rule to functions involving e^x.

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