derivative problem. is it same? First derivative of $y=\ln(x)^{\cos x}$ is $-\sin x\ln x+\frac{\cos x}{x}$ or another answer? My friend gets another answer, but it's true? thanks.

I will assume you mean $(\ln x)^{\cos x}$ and not $\ln(x^{\cos x})$.  I wish people would be explicit about that point.
If $g(x)$ is constant then $\dfrac d {dx} (f(x))^{g(x)} = g(x)f(x)^{g(x)-1}$.
If $f(x)$ is constant then $\dfrac d {dx} (g(x))^{g(x)} = f(g)^{g(x)} (\ln f(x)) f'(x)$.
If neither is constant then $\dfrac d {dx} (g(x))^{g(x)} = \text{the sum of those two things}$.
That can be proved by logarithmic differentiation.
