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)

Guida verificata passo dopo passo

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

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.

Video consigliato:

05:18

The Product Rule

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.

Video consigliato:

07:00

Taylor Polynomials

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.

Video consigliato: