Recently I came across a topic " total differential" which comes with a result d f = ∂ f ∂ x d x + ∂ f ∂ y d yAs much I learned in multivariable calculus this can be simplified as ∇ f ⋅ [ d x d y ] which graphically means taking directional Derivative in diraction of [ d x d y ] But is it Makes any sense ?

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Recently I came across a topic &quot; total differential&quot; which comes with a result  d  f  =            ∂      f              ∂      x        d  x  +            ∂      f              ∂      y        d  yAs much I learned in multivariable calculus this can be simplified as   ∇  f  ⋅      [                            d          x                                      d          y                      ]   which graphically means taking directional Derivative in diraction of       [                            d          x                                      d          y                      ]   But is it Makes any sense ?

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Observe that the derivative of a function   f  =  f  (  x  ,  y  ) is its gradient      g    r    a    d    f  =      [                                                                      ∂                f                                            ∂                x                                                                                                        ∂                f                                            ∂                y                                                          ]    .But if   f  (  x  ,  y  )  =  x (the projection in the x-axis) then its gradient is       [                                                      ∂              x                                      ∂              x                                                                          ∂              x                                      ∂              y                                            ]    =      [                            1                          0                      ]  andsimilarly for   f  (  x  ,  y  )  =  y (the projection in the y-axis), its gradient is       [                                                      ∂              y                                      ∂              x                                                                          ∂              y                                      ∂              y                                            ]    =      [                            0                          1                      ]  Hence the relation   d  f  =                    ∂        f                    ∂        x              d  x  +                    ∂        f                    ∂        y              d  y is the relation      [                                                                      ∂                f                                            ∂                x                                                                                                        ∂                f                                            ∂                y                                                          ]    =            ∂      f              ∂      x            [                            1                          0                      ]    +            ∂      f              ∂      y            [                            0                          1                      ]    ,where anyone clearly sees that   d  x  =      [                            1                          0                      ]   and   d  y  =      [                            0                          1                      ]   are the matrix forms of the derivatives of the projections onto both axis, that is, their gradients, respectively.

What is the direction of vector [[dx],[dy]]

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