Function can have two or more independent variables. To find derivative with respect to variable {partial derivative}|, hold other variables constant: df(x, y, z, ...) / dx = df(x, a, b, c, ...) / dx, where d is derivative, f is function, xyz are variables, and abc are constants.
change
For two variables, function-change slope or gradient df equals function value at x + dx and y + dx minus function value at x and y: df = f(x + dx, y + dy) - f(x,y) = (Df(x,y) / Dy) * dx + (Df(x,y) / Dy) * dy = D^2f(x,y) / DxDy, where d is differential, D is partial derivative, and D^2 is second partial derivative. Total change is sum of x and y changes. Direction change is partial derivative.
order
Order of taking partial derivatives does not matter, because variables are independent.
complex numbers
Complex-number differential depends on Cartesian differential. M is complex-number real part and N is imaginary part. dx and dy are infinitesimals on x-axis and y-axis. dq and dp are infinitesimals on real and imaginary axes. dq = M * dx + N * dy and dp = N * dx - M * dy. Therefore, Dp / Dx = Dq / Dy and Dp / Dy = - Dq / Dx {Cauchy-Riemann equations, partial derivative}, where D denotes partial differentials.
At point, two-variable function {implicit function, differentiation} can equal zero: f(x,y) = 0. Then, derivative of one variable by other variable, dy/dx, equals negative of function partial derivative with respect to x divided by function partial derivative with respect to y, because differentiation makes constant zero: dy/dx = -(Df(x,y) / Dx) / (Df(x,y) / Dy), where D is partial derivative.
Because differentiation makes constant zero, constant can be any value. Constant can be implicit-function-family {primitive function} parameter. Differential equation is sum of parameter equation and primitive function.
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Date Modified: 2022.0225