I've got two derivatives, one is logarithmic.

Okay, here is the first one:

f(x) = ln (x^2 * e^2x) / (x + ln x)

I tried doing this and the problem exploded into this huge mess. I know there is a simpler way and in fact this problem is simpler than most problems given I just forget how to simplify this. The best I could do was

Numerator can = 1 / (x^2 * e^2x)

Denominator can = x + 1/x

From this however, the entire thing explodes using the appropriate rules. Isn't there some way to simplify this further?

Okay phew, part II. Using logarithmic differentiation

f(x) = (cos(x))^x

lny = ln(cos(x))^x

1/y = 1/ x cos(x)

1/y = x * -sin(x) / cos(x)

I kinda block here. I must multiply by y on both sides but then the left side is just a "1".

I get y/y = y * -sin(x) * x / cos(x)

The y on the right can then be substituted by the original y but what happens on the left side?