Xy + sqrt[sin(6x)*ln(x)]?

Product rule and chain rule.

No, you need the chain rule first, since there's a composite function, √(sin(6x) * ln(x)), with a product inside.

What is f' cos6 supposed to mean? If this is supposed to mean the derivative of cos(6x), it's wrong.

For the 2nd line, I get what you're doing, but you're not using the notation well.
d/dx(ln(x)) = 1/x
That gets across the idea that you're differentiating ln(x), and that the derivative is 1/x.

Another way to say it is, if g(x) = ln(x), then g'(x) = 1/x.