I have a handle on calculating a derivative of $ f$ where $ f$ takes in an $ \mathbb{R}^2$ input and has an $ \mathbb{R}^1$ output (which would generally be $ \nabla f(x,y)$ ) and for there derivative of $ f$ where the input is in $ \mathbb{R}^1$ and the output is in $ \mathbb{R}^2$ (which would generally be $ \frac{dx(t)}{dt} \hat{i} + \frac{dy(t)}{dt} \hat{j} $ if $ x(t)$ and $ y(t)$ are the parameterization of $ f$ .

I’m trying to figure out what the dervative of $ f$ would be if it maps $ \mathbb{R}^2 \rightarrow \mathbb{R}^2$ .

As an example, let’s say $ f(x\hat i + y \hat j) = g(x,y) \hat i + h(x,y) \hat j$ . What would the sensible notion of a derivative be in this case? I feel like it would be $ \nabla f$ , but I’m also at a bit of a loss as to how to calculate that.