- #1
ak416
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Well actuallly 2 thms. They have to do with homogeneous functions. f(tx1,...,txn) = t^k * f(x1,...,xn). Now how do you show A) d/dx1 f(tx1,...,txn) = t^k-1 * d/dx1 f(x1,...,xn) and B) kt^(k-1)*f(x1,...,xn) = x1*d/dx1 f(tx1,...,xn) + xn*d/dxn f(x1,...,xn)
A) In the book They say that differentiating the first equation (definition of homogeneous function of degree k) by its first argument yields: d/dx1 f(tx1,...,txn) * t = t^k d/dx1 f(x1,...,xn) from which A easily follows. But how do they get this? I know you have to apply the chain rule somehow but I am not sure exactly...
B)Same as A, i end up with expressions like d/d(tx1) x1 which intuitively seems like t but I am not sure.
A) In the book They say that differentiating the first equation (definition of homogeneous function of degree k) by its first argument yields: d/dx1 f(tx1,...,txn) * t = t^k d/dx1 f(x1,...,xn) from which A easily follows. But how do they get this? I know you have to apply the chain rule somehow but I am not sure exactly...
B)Same as A, i end up with expressions like d/d(tx1) x1 which intuitively seems like t but I am not sure.