You do precisely what you've wrote down. let = real part of your function i.e. and let = imaginary part of your function i.e.

Now it's just straight-forward partial differentiation

Find . This means you need to differenciate g w.r.t x.

so =

and as is the partial derivative of h w.r.t y

we get

Clearly

A function is analytic in the complex plane if the BOTH caucy riemann equations are satisfied. So since the first C-R equation i checked is not satisfied you can say that the function is not analytic