Yes, you can simplify the expression \( \frac{\ln x + \log x}{x} \) to \( \frac{2 \log x}{x} \) if you're using natural logarithms (i.e., \( \log x = \ln x \)).

Let's clarify this:

If \( \log x \) and \( \ln x \) are both natural logarithms (which is common in many contexts), then \( \log x = \ln x \), and you can combine them:

\[
\frac{\ln x + \log x}{x} = \frac{\ln x + \ln x}{x} = \frac{2 \ln x}{x}
\]

Therefore, the derivative can be rewritten as:

\[
\frac{dy}{dx} = (\cos x)^x \left( \ln(\cos x) - x \tan x \right) + x^{\log x} \cdot \frac{2 \log x}{x}
\]

This simplifies to:

\[
\frac{dy}{dx} = (\cos x)^x \left( \ln(\cos x) - x \tan x \right) + 2 x^{\log x - 1} \log x
\]

This is a more compact form of the derivative, which incorporates the factor \( 2 \log x / x \) at the end.

Would you like to explore more about logarithmic differentiation or any other details?

Here are some related questions:

1. How do you differentiate a product of logarithmic and trigonometric functions?
2. What happens if \( \log x \) represents the logarithm in a different base?
3. Can you apply similar techniques to differentiate \( y = (e^x)^x \)?
4. How would the derivative change if \( y = x^{\sin x} + (\tan x)^x \)?
5. How does the derivative of \( y = x^{\log x} \) change if the base of the logarithm is not \( e \)?

**Tip:** Always ensure that the notation for logarithms is clear. In many contexts, \( \log x \) is used to denote the natural logarithm, but in others, it might refer to the base-10 logarithm.

Learn how to simplify \( \frac{\ln x + \log x}{x} \) to \( \frac{2 \log x}{x} \) and analyze its derivative. Understand the application of logarithmic differentiation and trigonometric functions in calculus. Suitable for advanced high school students and anyone studying calculus.

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