Short-Run Production counted the extra output from each added worker in a table. A production function $Q = f(K, L)$ says how much output any mix of capital $K$ and labor $L$ can make, and Partial Derivatives turns "the extra output from a little more labor" into an exact rate. The firm side of the course mirrors the consumer side: production functions play the role of utility functions, but their numbers are real output, not just rankings.
Key idea
$$MP_L = \frac{\partial Q}{\partial L}, \qquad MP_K = \frac{\partial Q}{\partial K}, \qquad AP_L = \frac{Q}{L}.$$ The marginal product is the extra output from a little more of one input, the others held fixed; the average product is output per unit of that input. Diminishing marginal product of labor means $\partial^2 Q/\partial L^2 < 0$.
Why it works
The marginal product is a slope: hold $K$ fixed, and $Q$ becomes a function of $L$ alone, whose derivative is $MP_L$. The average product is a different number, and the two are linked: $$\frac{d}{dL}\left(\frac{Q}{L}\right) = \frac{L\cdot MP_L - Q}{L^2} = \frac{MP_L - AP_L}{L}.$$ So the average rises while the marginal worker adds more than the average and falls once she adds less, the same way a new test score pulls your average up only if it beats your average.
Worked example: Marginal and average product
$Q = 10K^{1/2}L^{1/2}$. At $K = 16$, $L = 4$, find $MP_L$ and $AP_L$.
- $Q = 10(4)(2) = 80$.
- $MP_L = 5K^{1/2}L^{-1/2} = 5\cdot\dfrac{4}{2} = 10$.
- $AP_L = 80/4 = 20$.
- The fifth worker adds about 10 units, half the average. For Cobb-Douglas, $MP_L = \beta\cdot AP_L$, with $\beta = \tfrac{1}{2}$ here.
Answer: $MP_L = 10$, $AP_L = 20$.
Worked example: A short-run production function
With its building fixed, a bakery makes $Q = 20L + 3L^2 - 0.1L^3$. Find $MP_L$ and $AP_L$ at $L = 10$, and where diminishing marginal returns begin.
- $MP_L = 20 + 6L - 0.3L^2 = 20 + 60 - 30 = 50$.
- $AP_L = 20 + 3L - 0.1L^2 = 20 + 30 - 10 = 40$.
- $MP_L$ peaks where $6 - 0.6L = 0$: $L = 10$. Beyond 10 workers each new worker adds less than the last, though output keeps rising until $MP_L$ reaches zero.
Answer: $MP_L = 50$, $AP_L = 40$; diminishing returns set in after $L = 10$.
Common mistake
Treating the marginal product as output per worker. Output per worker is the average product. In the first example the average is 20 while the next worker adds only 10, and a firm deciding whether to hire needs the 10. Check: "marginal" always means a derivative (or the change from one more unit), never a ratio of totals.