mathematics//calculus//partial derivative

A partial derivative is the derivative of a function of several variables with respect to one of them while the others are held fixed, and it is how an engineer asks a sensitivity question of a system with many inputs: how much does this output move if I change only that input. The flow out of a tank's orifice depends on the level and on the valve opening; the partial derivative with respect to the level says how much the outflow rises per centimetre of level at a fixed opening, and the one with respect to the opening says the converse at a fixed level. Each is an ordinary derivative taken along one axis, written with a curly \(\partial\):

A partial derivative is the derivative of a function of several variables with respect to one of them while the others are held fixed, and it is how an engineer asks a sensitivity question of a system with many inputs: how much does this output move if I change only that input. The flow out of a tank's orifice depends on the level and on the valve opening; the partial derivative with respect to the level says how much the outflow rises per centimetre of level at a fixed opening, and the one with respect to the opening says the converse at a fixed level. Each is an ordinary derivative taken along one axis, written with a curly ∂\partial∂:

∂f∂xj(a)=lim⁡h→0f(a+h ej)−f(a)h,\frac{\partial f}{\partial x_j}(a)=\lim_{h\to 0}\frac{f(a+h\,e_j)-f(a)}{h},∂xj​∂f​(a)=h→0lim​hf(a+hej​)−f(a)​,

where eje_jej​ moves only the jjj-th input. The value depends on the point aaa, and on the other inputs having been frozen there.

Partial derivatives are the raw material of every multivariable tool. Stacked as a vector for a scalar function they form the gradient, the direction of steepest increase (gradient, divergence and curl); arranged as a matrix for a vector function they form the Jacobian, the object that linearization and the extended Kalman filter compute; taken twice they form the Hessian.

Holding the others fixed is a modelling choice, and in a coupled plant it is often an impossible experiment. Raising a pump's speed also raises the pressure that the downstream flow depends on, so the partial derivative of flow with respect to speed (pressure frozen) is different from the total change seen on the plant. The chain rule adds the indirect paths back.

An equation with partial derivatives in time and in space (temperature along a bar, stress in a beam) is a PDE, as opposed to an ODE, whose only variable of differentiation is time. In practice a PDE is chopped into nodes and becomes a large system of ODEs.