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# Partial derivatives

When strolling along a path – 1 dimension – through the hilly landscape, the slope of the path clearly tells how steep the hill is along that path. The derivative in that direction tells how steep the hill is in that direction only.

When walking on a hill with no path laid out, with the freedom to go anywhere in the grass – 2 dimensions – there is not just one but infinitely many directions to walk in.

• Walk one walk and you can find the derivative which gives the slope, the steepness, in that direction.
• Walk another way and you’ll find another derivative giving another slope, another steepness.
• You can walk in infinitely many different directions by turning just a little each time, and each will have a new derivative.

Clearly, none of these derivatives represents the point on the hill in full – they each represent a part of the point on the hill. Let’s call them partial derivatives.

As we remember from the Dimensions skill in the Existence discipline, we only need two directions in a 2-dimensional space to cover all possible directions. (That’s why two coordinate axes are enough.) Thus, two partial derivatives are enough to represent all partial derivatives on a 2D surface. We can for instance pick those that point along the $x$ and $y$ axes:

$$s’_x\qquad\text{ and }\qquad s’_y$$

When you hear someone say ‘partial derivatives’, it is typically these two (no more, no less) they are referring to.[1,2]

References:

1. Introduction to partial derivatives’ (web page), Duane Q. Nykamp, Math Insight, www.mathinsight.org/partial_derivative_introduction (accessed Jun. 14th, 2020)
2. Funktioner af to variable’ (web page, in Danish), Karsten Schmidt, Technical University of Denmark, Institute for Mathematics and Computer Science, eNote 19, 2016, 01005.compute.dtu.dk/enotes/19_-_Funktioner_af_to_variable (accessed Jun. 14th, 2020), chapter 19