Multivariate Calculus Help!

[raininegypt]

Here's the homework problem:

A manufacturer estimates that production (in hundreds of units) is a function of the respective amounts x and y of labor and capital used as follows:

f(x,y)= [1/3x^-1/3 + 2/3y^-1/3]^-3
Find and interpret fx(27,64) and fy(27,64)

[FONT=verdana, geneva, lucida, lucida grande, arial, helvetica, sans-serif]I've tried distributing the -3 exponent but I didn't get too far because it left me without the proper variables to solve. I have looked though example after example and have had no luck finding out how to solve this problem[/FONT]

 

[stapel]

A manufacturer estimates that production (in hundreds of units) is a function of the respective amounts x and y of labor and capital used as follows:

f(x,y)= [1/3x^-1/3 + 2/3y^-1/3]^-3
Find and interpret fx(27,64) and fy(27,64)

I've tried distributing the -3 exponent but I didn't get too far because it left me without the proper variables to solve.

What do you mean by "distributing" the exponent? What do you mean by "proper variables"?

When you reply, please include a clear listing of your efforts so far. Thank you!

 

[Ishuda]

Here's the homework problem:

A manufacturer estimates that production (in hundreds of units) is a function of the respective amounts x and y of labor and capital used as follows:

f(x,y)= [1/3x^-1/3 + 2/3y^-1/3]^-3
Find and interpret fx(27,64) and fy(27,64)

I've tried distributing the -3 exponent but I didn't get too far because it left me without the proper variables to solve. I have looked though example after example and have had no luck finding out how to solve this problem

[g^t(x,y)]_x = t g^t-1(x,y) g_x(x,y)
[g^t(x,y)]_y = t g^t-1(x,y) g_y(x,y)

 

[raininegypt]

I multiplied the -3 exponent with the other two exponents

 

[stapel]

I multiplied the -3 exponent with the other two exponents

Exponents do not "distribute" over addition or subtraction. (They were supposed to have covered that back in algebra.) Instead, you'd need to invert the expression, and then cube it, if you wish to expand it.

. . . . .\(\displaystyle \left[\dfrac{1}{3x^{-\frac{1}{3}}}\, +\, \dfrac{2}{3y^{-\frac{1}{3}}}\right]^{-3}\)


. . . . .\(\displaystyle \left[\dfrac{x^{\frac{1}{3}}}{3}\, +\, \dfrac{2y^{\frac{1}{3}}}{3}\right]^{-3}\)


. . . . .\(\displaystyle \left[\dfrac{x^{\frac{1}{3}}\, +\, 2y^{\frac{1}{3}}}{3}\right]^{-3}\)


. . . . .\(\displaystyle \left[\dfrac{3}{x^{\frac{1}{3}}\, +\, 2y^{\frac{1}{3}}}\right]^{3}\)


. . . . .\(\displaystyle \left[\dfrac{3}{x^{\frac{1}{3}}\, +\, 2y^{\frac{1}{3}}}\right]\, \cdot\, \left[\dfrac{3}{x^{\frac{1}{3}}\, +\, 2y^{\frac{1}{3}}}\right]\, \cdot\, \left[\dfrac{3}{x^{\frac{1}{3}}\, +\, 2y^{\frac{1}{3}}}\right]\)


...and so forth.