La tasa de cambio del beneficio (o ganancia) en la producción es la derivada del beneficio con respecto a la cantidad producida, es decir, el beneficio marginal \( P'(x) \).

Dado que:

\[
P(x) = 60x + 0.01x^2 - 20,000
\]

La tasa de cambio del beneficio con respecto a \( x \) es:

\[
P'(x) = 60 + 0.02x
\]

Luego, evaluamos esta tasa de cambio en \( x = 3100 \) y \( x = 3200 \) toneladas:

- Para \( x = 3100 \):
  \[
  P'(3100) = 60 + 0.02(3100) = 60 + 62 = 122 \text{ dólares por tonelada}
  \]

- Para

Explore how to calculate marginal profit using derivative concepts with a detailed example. Learn about the profit function P(x) = 60x + 0.01x^2 - 20,000 and its derivative P'(x) = 60 + 0.02x. Suitable for advanced high school students studying calculus.

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