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Say we have function f(x,y) that has partial derivatives at (a,b).
Then in general, is the following true (say, for f_x(a,b) ):
1) f_x(a,b) = d/dx [ f(x, b) ] @x=a
2) first sub y = b into function f_x(x,y) ,
then sub x=a into f_x(x, b) , does the result always equal f_x(a,b)?
[assuming f_x(x,b) is a finite defined value @x=a]
Example of where I might use this:
f(x,y) = e^(x^3 + y^3)^(1/3) *defined at (0,0)
f_x(x,y) = x^2 * e^(x^3 + y^3)^(1/3)
========================== *not defined at (0,0)
(x^3 + y^3)^(2/3)
this function gives 0/0 when attempting to find value at (0,0), but
using both methods we get correct result:
1) f_x(0,0) = d/dx [ f(x,0) ] @x=0
= d/dx [e^(x^3)^(1/3)] @x=0
= d/dx [e^x] @x=0
= e^x @x=0
= 1
2) plug y=0 into f_x(x,y)
f(x,0) = x^2 * e^(x^3)^(1/3)
==================
(x^3)^(2/3)
x^2 * e^x
= ================
x^2
= e^x = f(x,0)
therefore f(0,0) = f(x=0, 0) = e^0 = 1
Do both these methods work in general (if the partials do exist by
limit def. and result of methods mentioned are finite defined values) ?
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