The formula
∂f/∂x: differentiate with respect to x holding the other variables constant
Why it works
Partial differentiation involves no new rule: it is ordinary differentiation with every variable but one treated as a number. What changes is the interpretation. A function of two variables is a surface, and each partial is the slope of that surface in one direction: the x one looking east, the y one looking north. The gradient collects them into a vector pointing where the function grows fastest, and its magnitude says how fast. Where the gradient vanishes there is a critical point: a maximum, a minimum or a saddle.
How to solve it by hand
- Pick the variable you are differentiating with respect to
- Treat every other one as a constant and differentiate as usual
- Repeat for each variable to get the whole gradient
- Evaluate at the point if you want the actual slopes
What is worth knowing
Mixed derivatives — differentiating first by x and then by y, or the other way round — give the same thing whenever they are continuous, and that is Schwarz’s theorem. Checking it is the best way to catch an arithmetic slip, because if they disagree something has almost certainly been differentiated wrong. One thing people forget: a vanishing gradient is not enough to tell a maximum from a minimum. You have to look at the second derivatives, and a function like x² − y² has zero gradient at the origin without being either: it is a saddle, rising in one direction and falling in the other.