00:01
The discussion says that the productivity of a country is given by f x comma y is equal to 13 .89 into x to the power 0 .4 to 3 into y to the power 0 .577 where x is the amount of labor and y is the amount of capital.
00:20
In part we are asked to find the partial differentiation of f with respect to x and with respect to y.
00:28
In part v we are asked to find the value of partial differentiation of f with respect to x at the point 66 .58 and the value of partial differentiation of f with respect to y at the point 66 .58.
00:45
So let's start to solve these problems.
00:48
Here first we will find the partial differentiation of f with respect to x.
00:52
Partial differentiation of f with respect to x denoted by f subx is equal to dahl over dahl x of 13 .89 x to the power 0 .423, y to the power 0 .577.
01:10
When we take partial differentiation with respect to x, then variables then x give like a constant.
01:17
That means here, this term, behaves like a constant.
01:22
13 .89, y to the power 0 .577, dahl over dahl x of x to the power 0 .423.
01:34
13 .89y to the power 0 .577.
01:40
Derivative of this with respect to x is equal to 0 .423 x to the power 0 .423 minus 1.
01:51
By simplifying this, we obtain f subx is equal to 5 .87 -547 y to the power 0 .57 into x to the power negative 0 .577.
02:12
F sub x is equal to 5 .8757.
02:19
Y over x to the power 0 .577.
02:23
Next we will find the partial definition of f with respect to y.
02:29
Partial definition of f with respect to y denoted by f sub y is equal to del over del y of 13 .89 x to the power 0 .4 to 3, y to the power 0 .577.
02:45
13 .89 x to the power 0 .4 to 3.
02:50
Del over del y of y of y to the power 0 .577.
02:58
13 .89 x to the power 0 .423 derivative of this with respect to y of y of y of y.
03:05
Is equal 0 .577, y to the power 0 .577 minus 1, is equal to 8 .01, 453, x to the power 0 .423, y to the power, power negative, 0 .4 to 3.
03:31
Ab sub y is equal to 8 .01, 453, x over y to the power, 0 .422...