... equals arctan (y/x).

arctan(y/x) - Wolfram|Alpha

Notice the partial derivative wrt x. Click on 'show steps'. Just in case a picture helps...

... where (key in spoiler) ...

Spoiler:

... equals a vector composed of both partial derivatives:gradient of delta(x,y)?

Gradient - Wikipedia, the free encyclopedia

gradient of arctan(y/x) - Wolfram|Alpha

.. as for any vector: the length or magnitude using pythagoras and the direction relative to the x-axis using arctan as we did above for delta.determine the length and direction

For length, simplify

For direction, arctan of a simplification of the 'y' component over the 'x'.

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Balloon Calculus; standard integrals, derivatives and methods

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