Multivariable Instantaneous rate of change clarificationWhen you are computing the instantaneous rate of change for f(x,y) what do you take the derivative with respect to?for example, forf(x,y)=(sin(πx)cos(πy),yexy,x2+y3)If I was to find the instantaneous rate of change for all 3 of these functions going through (1,2) with the velocity vector (3,-2) would I just take ddx of all of the functions at (3,-2)?

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HadoHaurrysap3w · Accepted Answer

If you want the change in f as both x and y are changing, then you are going to need something that relates x to y. (a parametric curve perhaps, or a direction vector).Without that, you can evaluate the sentivity of f to changes in x this is the partial derivative.             ∂      f              ∂      x      And there is also a partial derivative with respect to y, i.e.             ∂      f              ∂      y      If you have the partials, and a parametric curve then you can find the &quot;total derivative&quot;            d      f              d      t        =            ∂      f              ∂      x                  d      x              d      t        +            ∂      f              ∂      y                  d      y              d      t       and these will correspond to the same derivative you would get if you subsituted x(t),y(t) into your function.Since f is a vector, you will get a derivatives for each of the i,j,k components.

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