APFIVE In a study to determine the optimal number of units to sell for maximum profit, a researcher analyzed the profit function $$P(x)=\ln(x)-0.1*x$$, where $$x$$ represents thousands of units sold. To find the maximum profit, the researcher differentiated $$P(x)$$ but erroneously computed the derivative of $$\ln(x)$$ as $$\ln(x)$$ instead of the correct derivative $$\frac{1}{x}$$. What mistake did the researcher commit?
They applied the product rule where it was not required, complicating the derivative unnecessarily.
They misapplied the derivative of $$\ln(x)$$, confusing it with the function itself instead of using $$\frac{1}{x}$$ as the correct derivative.
They incorrectly differentiated the linear term by introducing an extra constant.
They failed to differentiate the $$-0.1*x$$ term entirely, leading to an incomplete derivative.
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