Partial Derivative
A manufacturer estimates that the annual output at a certain factory is given by Q(K,L)=(42K^0.8)(L^0.2) units, where K is the capital expenditure in thousands of dollars and L is the size of the labor force in worker-hours. Suppose that the current capital expenditure is $240000 and the labor level is 560 worker-hours.

Question

Partial Derivative 
A manufacturer estimates that the annual output at a certain factory is given by Q(K,L)=(42K^0.8)(L^0.2) units, where K is the capital expenditure in thousands of dollars and L is the size of the labor force in worker-hours. Suppose that the current capital expenditure is $240000 and the labor level is 560 worker-hours.

tkhunny · Answer

Where is the partial derivative?  So far, just substitute the given values.

anonymous · Answer

I tried that but it did not work. And our homework is over partial derivatives so I was unsure what we were supposed to do lol

anonymous · Answer

Find the current marginal productivity of capital QK

tkhunny · Answer

Aha!!  A question.  That we can do.  Have you considered the Total Derivative?

anonymous · Answer

Sorry I forgot that part haha...
No I have not, should I try it? (33.6K^-.2)(.2L^-.8) is the derivative, right?

anonymous · Answer

I tried it and it didn't work

tkhunny · Answer

In one variable.

$A(t) = \pi r^{2}$

$\dfrac{dA}{dr} = 2\pi r$ or the differential form $dA = 2\pi r\;dr$

This is all we are doing.  We are just doing it in multiple dimensions.  A partial deriviative re

Q(K,L)=(42K^0.8)(L^0.2)

$\dfrac{\partial Q}{\partial K} = 48\cdot 0.8\cdot K^{0.8 - 1.0}\cdot L^{0.2}$

or the differential form.

$\partial Q = \left(48\cdot 0.8\cdot K^{0.8 - 1.0}\cdot L^{0.2}\right)\;\partial K$

Do that for L and add them up.

anonymous · Answer

So do I just plug in 240,000 for K and 560 for L?

tkhunny · Answer

1) "Plug In" doesn't actually mean anything.

2) No.  Just above, you were asked to find $\dfrac{dQ}{dL}$.  Did you do that?

anonymous · Answer

No...

tkhunny · Answer

Well...

anonymous · Answer

I'm not sure what I'm supposed to do lol

tkhunny · Answer

You are finding a partial derivative with respect to L.  Treat K as a constant and find the 1st Derivative.

anonymous · Answer

2(-.16(K)^(-1.2))(L^.2)?

tkhunny · Answer

Why did the K exponent change?  That's no good.  Treat K like a constant.  Just carry it along for the ride.

Why did you add 1 to the L exponent.  This is a derivative.  It should go DOWN by 1.

anonymous · Answer

(.8K^-.2)(.2L^-.8)2K

anonymous · Answer

?

tkhunny · Answer

(42K^0.8)(L^0.2) 

Change Nothing: (42K^0.8)
Find the Derivative: (L^0.2)