关于 x 与 y 的偏导数计算器

A multivariable function can change in several independent directions. The notation ∂f/∂x asks one precise question: how does f change when x varies and every other independent variable is held fixed? For ∂f/∂y, the roles reverse. This instruction is what separates a partial derivative from an ordinary one-variable derivative, even though the algebraic rules look familiar. Before differentiating, circle the variable named in the operator. Treat any other independent symbol as a constant coefficient. For f(x,y)=x²y+y³, differentiating with respect to x turns x²y into 2xy because y is fixed, while y³ becomes zero. Differentiating with respect to y turns x²y into x² and y³ into 3y². The two correct answers differ because they measure different directional changes on the same surface. Product and chain rules still apply. For f=sin(xy), the inside is xy. With respect to x, y is constant, so the inner derivative is y and the result is y cos(xy). With respect to y, the inner derivative is x, producing x cos(xy). For exp(x²+y²), the x partial is 2x exp(x²+y²), while the y partial is 2y exp(x²+y²). Keeping the unchanged inner expression visible is a useful check. Higher-order notation records repeated directions. The expression ∂²f/∂x² means differentiate twice with respect to x. A mixed partial such as ∂²f/∂y∂x applies the operations in the indicated order. On this page, repeated partials for one selected variable use the order control. For a mixed derivative, calculate the first partial, copy its result, switch the variable, and calculate again. That two-pass workflow keeps the order explicit and reduces notation mistakes. Geometrically, z=f(x,y) is a surface. Fixing y gives a curve cut through the surface in the x direction, and ∂f/∂x is the slope of that curve. Fixing x gives the perpendicular y-direction slice. At a point, these two slopes form the components used to build the gradient. They do not by themselves describe every directional derivative, but they are the foundation for optimization, tangent planes, and multivariable models. Units matter in applications. If f is temperature and x is distance, ∂f/∂x is a temperature change per unit distance while the other coordinates remain fixed. If f is cost depending on quantity and time, the chosen partial describes one marginal effect under a stated hold-fixed assumption. Always carry those assumptions into the interpretation; a symbolic expression alone cannot tell you whether the variables are truly independent in the real system. Check a result by asking whether all inactive-variable-only terms disappeared and whether inactive symbols remained as constants where they multiply active terms. You can also compare the x and y results for symmetry when the original function is symmetric. CalculusMate uses the same deterministic differentiation tree as the ordinary derivative tool, with the selected symbol defining what changes. Use the implicit differentiation calculator instead when x and y are linked by an equation and y must follow x rather than remain fixed.