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.