We define marginal product of labor, MP_L as the derivative of f with respect to the L - which is, as (approximately) how much Y will increase when L increases by one unit. We also describe the marginal product of capital, MP_K as the derivative of f with respect to K. Note that MP_L and MP_K will rely on both L and K (MP_L and MP_K are functions, not variables). 

MP_L = df/dL, MP_K= df/dK

• since f is increasing in L, MP_Lshould be positive for all values of L and K.
• MP_Lpresumed to be decreasing in L - the more work that is used, the lower the marginal product of labor.
• MP_Lpresumed to be increasing in K - the more capital, the higher the marginal product of labor.
• Similarly MP_Kshould be positive for all values of L and K.
• MP_K is presumed to be decreasing in K and increasing in L.

When we view Y as a function of L holding K fix, Y will be increasing in L however at a decreasing pace (Because of the fact that MP_L is positive though decreasing in L). 

Figure:Production function

We state labour productivity as Y/L that is, as GDP per hour worked. Labour productivity tells us how much we can produce by employing one hour of labour and it relies on the amount of capital and the technology.