For f(x) near x = a, f(a + h) - f(a) = (h/c) * C + (h / (2*c)) * (h/c - 1) * C^2 + ..., where C = f(a + c) - f(a), c = x change, and h = x - a {method of finite differences} {finite differences method}.
Rules can find nth derivative of function product {Leibniz's theorem} {Leibniz theorem}. If w = u*v, (D^n)w = (D^n)u * v + ... + (D^(n/2))u * (D^(n/2))v + ... + u * (D^n)v, where D^n is nth partial derivative and D^(n/2) is (n/2)th partial derivative. The rule makes binomial expansion series with n + 1 terms.
Limit of function ratio is limit of function first-derivative ratio, if function limit equals zero or if denominator-function limit is positive infinity or negative infinity {L'Hôspital's rules} {L'Hôspital rules}.
For functions of functions {function of function differentiation}, multiply differential of main function and differential of other function: dg(f) = (dg / df) * df {chain rule, differentiation}. Other function can be independent variable: df(x) = (df(x) / dx) * dx.
Derivative of two-variable function f(x,y) with respect to variable t is (Df(x,y) / Dx) * (dx/dt) + (Df(x,y) / Dy) * (dy/dt), where D is partial derivative {chain rule, partial derivatives}.
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Date Modified: 2022.0225