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To whom it may concern,
I have been struggling to understand the chain rule on PARTIAL differentiation. With an example of a function f(u,v) where u(x,y) and v(x,y),
why is fx = fu*ux + fv*vx please??.... why '+' but not multiplication or division or substraction?
where
fx = df/dx (in partial derivatives)
fu = df/du (in partial derivatives)
ux=du/dx (in partial derivatives) and similar for fv and vx.
i look forward to hear and discuss thank you.
Regards,
tmnt007
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You add because that's what you do with dot products...
The chain rule in one-dimension is given as:
The equivalent chain-rule for f(r(t)) where r = x(t)i + y(t)j can be written:
where
And the dot product of these vectors gives us our scalar derivative:
Now, to find the partial derivative of f(u,v) with u = u(x,y), v = v(x,y) with respect to x, use the above rule but keep y fixed and differentiate in the usual way to get:
...so the reason you add, instead of multiplying or anything else, is simply because the chain rule with more than one variable arises from a dot product.
Last edited by Dross (2006-09-05 08:58:52)
Bad speling makes me [sic]
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