Completing partial derivatives to make them converge

For a function $f(x,y)$ of two independent variables we have an incomplete specification of its partial derivatives as follows:

$\frac{\mathrm{\partial}f(x,y)}{\mathrm{\partial}x}=\frac{1}{g(x,y)\sqrt{1-(\frac{ky}{{x}^{(1/3)}}{)}^{2}}}$

$\frac{\mathrm{\partial}f(x,y)}{\mathrm{\partial}y}=\left(\frac{3x}{4}\right){\left(\frac{k}{{x}^{(1/3)}}\right)}^{2}(2y)\frac{1}{g(x,y)\sqrt{1-(\frac{ky}{{x}^{(1/3)}}{)}^{2}}}$

Problem: finding a suitable $g(x,y)$ that makes the partial derivatives converge to a single function $f(x,y)$ that fulfills the condition $f(x,0)=x$

I will be grateful if people with many flight hours can offer suggestions for $g(x,y)$. Needless to say, I am not asking that they verify those suggestions, but in case someone would like, these are the inputs to Wolfram integrator:

$\frac{1}{(g(x,y\text{as}r)}\sqrt{1-(kr/{x}^{(}1/3){)}^{2})}\phantom{\rule{0ex}{0ex}}\frac{(3t/4)(k/{t}^{1/3}{)}^{2}(2x)}{(g(x\text{as}t,y\text{as}x)\sqrt{1-(kx/{t}^{(}1/3){)}^{2})}}$

For a function $f(x,y)$ of two independent variables we have an incomplete specification of its partial derivatives as follows:

$\frac{\mathrm{\partial}f(x,y)}{\mathrm{\partial}x}=\frac{1}{g(x,y)\sqrt{1-(\frac{ky}{{x}^{(1/3)}}{)}^{2}}}$

$\frac{\mathrm{\partial}f(x,y)}{\mathrm{\partial}y}=\left(\frac{3x}{4}\right){\left(\frac{k}{{x}^{(1/3)}}\right)}^{2}(2y)\frac{1}{g(x,y)\sqrt{1-(\frac{ky}{{x}^{(1/3)}}{)}^{2}}}$

Problem: finding a suitable $g(x,y)$ that makes the partial derivatives converge to a single function $f(x,y)$ that fulfills the condition $f(x,0)=x$

I will be grateful if people with many flight hours can offer suggestions for $g(x,y)$. Needless to say, I am not asking that they verify those suggestions, but in case someone would like, these are the inputs to Wolfram integrator:

$\frac{1}{(g(x,y\text{as}r)}\sqrt{1-(kr/{x}^{(}1/3){)}^{2})}\phantom{\rule{0ex}{0ex}}\frac{(3t/4)(k/{t}^{1/3}{)}^{2}(2x)}{(g(x\text{as}t,y\text{as}x)\sqrt{1-(kx/{t}^{(}1/3){)}^{2})}}$