How to solve
A mixed partial differentiates once in each variable in turn. For smooth functions (including polynomials), Clairaut’s theorem says the order does not matter: f_xy = f_yx.
Steps
- Differentiate with respect to x, treating y as constant (or start with y).
- Differentiate that result with respect to the other variable.
- For these polynomial drills, either order is fine.
Example
If f(x, y) = x²y³, find f_xy.
- f_x = 2xy³.
- Differentiate in y: f_xy = 2x · 3y² = 6xy².
Answer: 6xy^2
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