Consider the graph of $f(x,y) = \sqrt{xy}$. Each coloured line depicts the curve $f(x,y) = k$ for different values of $k \in \mathbb{R}$.
The derivative is thought as the tangent. How do I think of the partial derivatives $\frac{\delta f}{\delta x}$ and $\frac{\delta f}{\delta y}$ in the graph?
The partial derivatives tell us how the function changes when only one variable is changed. For example, $\frac{\partial f}{\partial x}$ says how the value of $f$ will change if we increase $x$.
We can make sense of that in your graph. We fix a $y$ value and look at what happens as we move to the right along the $x$ axis. We can see that at $y=4$, $\frac{\partial f}{\partial x}$ is greater than at $y=2$ because the $k$ lines are closer together,