Math Is Fun Forum

tmnt007

Member

Registered: 2006-09-05

Posts: 1

chain rule on partial differentiation

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

Dross

Member

Registered: 2006-08-24

Posts: 325

Re: chain rule on partial differentiation

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)

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