Does a function need to have a parameter underlying all inputs to be totally differentiable?

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I have the following function $$e(p_x,p_y,\bar{U}) = p_x a(p_x,p_y,\bar{U}) + p_y b(p_x,p_y,\bar{U})$$

Can I take the total derivative of this? I am confused about the criteria for something to be totally differentiable.

My Question:

  1. Is the function above totally differentiable (assuming the functions chosen aren't weird or kinky or something)?
  2. How can I tell if something is totally differentiable?