Interpreting Partial Derivatives

Interpreting a partial derivative simply means reading it as a slope or a rate: "if I step this way, how fast does the thing I'm measuring go up or down?" It's how an engineer reads a temperature map, or an economist reads a production model. The formula $f_x(a, b) = \frac{d}{dx}f(x, b)\big|_{x = a}$ looks fussy, but it's just three steps: plug in $y = b$ (freeze the north-south position), which leaves $f(x, b)$ (a curve in $x$ alone), then take its ordinary derivative at $x = a$ (the slope there). That's all a partial derivative is.

Two slices. The last lesson computed partials by holding a variable fixed. Here's what that looks like. Stand at the point $(a, b, f(a, b))$ on the surface. Holding $y = b$ fixed is the same as slicing the surface with the vertical plane $y = b$, which cuts out a curve, and $f_x(a, b)$ is that curve's slope. Slice the other way, with the plane $x = a$, and you get another curve whose slope is $f_y(a, b)$. Notice the two can be completely different: a hillside can be steep going north and nearly flat going east.

Rates. Like any derivative, a partial has units of "output per input." If $T(x, y)$ is temperature in °C and $x, y$ are in meters, then $T_x$ is in °C per meter: how fast it warms as you walk east.

Economics. For production $P(L, K)$ from labor and capital, $P_L$ is the marginal product of labor: about how much extra output you get from one more worker, with capital held fixed. Holding the other input fixed is exactly what makes the question fair.

A common mistake. It's tempting to treat $\frac{\partial P}{\partial V}$ like a fraction and cancel, since $\frac{dy}{dx}$ often behaves like one. It isn't one. Each partial derivative holds a different variable fixed, so you can't cancel them like fractions. There's a challenge problem below where canceling would predict $+1$ and the true answer is $-1$.

Sensitivity analysis asks which input matters most: the partial derivatives of a model's output with respect to each input rank them.