Pregunta
Find (a) \( \frac{d}{d x}\left(x^{3} e^{x}\right) \)
Ask by Hall Rojas. in the United States
Feb 25,2025
Solución de inteligencia artificial de Upstudy
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The derivative of \( x^{3} e^{x} \) with respect to x is \( x^{2} e^{x} (3 + x) \).
Solución
We are given the function f(x) = x³eˣ. To find its derivative, we use the product rule, which states that if f(x) = u(x)v(x), then
f '(x) = u '(x)v(x) + u(x)v '(x).
Let u(x) = x³ and v(x) = eˣ.
Step 1: Differentiate u(x) and v(x):
u '(x) = 3x²,
v '(x) = eˣ (since the derivative of eˣ is eˣ).
Step 2: Apply the product rule:
f '(x) = (3x²)(eˣ) + (x³)(eˣ).
Combine like terms:
f '(x) = eˣ(3x² + x³).
You can also factor x² from the expression:
f '(x) = x²eˣ(3 + x).
Thus, the derivative is:
d/dx (x³eˣ) = x²eˣ(3 + x).
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Simplifique esta solución Extra Insights
To differentiate \( x^{3} e^{x} \) using the product rule, we let \( u = x^{3} \) and \( v = e^{x} \). According to the product rule, \( \frac{d}{dx}(uv) = u'v + uv' \). First, we need to find the derivatives: - \( u' = \frac{d}{dx}(x^{3}) = 3x^{2} \) - \( v' = \frac{d}{dx}(e^{x}) = e^{x} \) Now, applying the product rule: \[ \frac{d}{dx}(x^{3} e^{x}) = u'v + uv' = (3x^{2})(e^{x}) + (x^{3})(e^{x}) \] We can factor out \( e^{x} \): \[ = e^{x}(3x^{2} + x^{3}) \] Thus, the derivative is: \[ \frac{d}{dx}(x^{3} e^{x}) = e^{x}(3x^{2} + x^{3}) \]
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