I am trying to calculate the mean curvature of a curve parametrized by $( x(t), y(t))$ in a two-dimensional space with ambient metric $g$. I know one formula for the mean curvature, $K = \nabla_i \hat{n}^i$ where $\nabla_i$ is the covariant derivative with respect to $g$ and $\hat{n}^i$ is the unit normal vector. I can write $\hat{n}^i$ in terms of the functions $x(t)$ and $y(t)$, and their first derivatives. I get stuck trying to calculate things like $\partial_x \hat{n}^x$. What is $\partial_x x'(t)$? Can I naively take this to equal $x''(t) / x'(t)$? If I do so, I seem to get the wrong result, but I am not sure if I am making a mistake or I have an incorrect understanding.
To be concrete, I am taking the hyperbolic metric $g_{xx} = g_{yy} = 1/y^2$. I can then write the normal vector as $$ \hat{n} = \frac{y}{\sqrt{x'^2 + y'^2}} (y', -x') $$ Thank you for taking the time to answer this. I am trying to calculate the extrinsic curvature term $K$ in the Gibbons-Hawking boundary term.