For continuous functions, range has change rate {derivative, function} {first derivative} with domain {differentiation, mathematics}. Functions f have independent-variable domain, such as time t, and dependent-variable range, such as distance x: f(t) = x. You can differentiate to find how distance x varies with time t: velocity v = df = dx / dt. For functions whose domain is time and whose range is velocity, you can differentiate to find how velocity v varies with time t: acceleration a = dv / dt.
Curve or surface functions have y-axis range and x-axis domain. At domain and range points (x,y), you can differentiate to find angle A of tangent to curve: tan(A) = dy / dx. You can also calculate slope of line normal to curve or surface.
See Figure 1. For function y = f(x) and two function points (x1, y1) and (x2, y2), change rate {slope, function} between two points is (y2 - y1) / (x2 - x1). Slope converges to value {limiting value} {limit} as x2 - x1 approaches zero at point (x1, y1). Limit is change rate at point (x1, y1).
See Figure 2. Functions have maximum or minimum where tangent slope is zero, because function has reached top or bottom.
At a point, if two functions have limits, function-sum limit is sum of function limits {limit theorem}. Function-product limit is product of function limits. Function-quotient limit is quotient of function limits. For functions, nth-root limit is nth root of limit.
Functions {non-decreasing function} can have derivatives >= 0 over intervals.
Functions {non-increasing function} can have derivatives <= 0 over intervals.
Functions can have derivatives {second derivative} of first derivatives. Second differential is d2x [2 is superscript] or d2f(x): (d^2)x or (d^2)f(x).
For discontinuous functions, operator {difference operator} can find finite difference between (n+1)th term and nth term: y(n + 1) - y(n). First-order difference operator symbol is uppercase Greek letter delta. Second-order difference operator symbol is uppercase Greek letter delta squared.
Vanishingly small increment or infinitesimally small interval can have symbol dx {differential}. dx = (x + delta_x) - x, as delta_x approaches zero. For function, df(x) = f(x + delta_x) - f(x), as delta_x approaches zero. Therefore, (f(x + dx) - f(x)) / ((x + dx) - x) ~ df(x) / dx or f'(x) and f(x + dx) ~ f(x) + f'(x) * dx ~ f(x) + df(x).
Variable partial derivative times variable differential is variable change {exact differential}: (Df(x,y) / Dx) * dx = (x change), where f(x,y) is a two-variable function, D is partial derivative, and dx is differential. For all variables, sum exact differentials. For two variables, (Df(x,y) / Dx) * dx + (Df(x,y) / Dy) * dy = (x change) + (y change). First-order differential equation can use differentials.
At point, in direction, surface has slope {directional derivative}|. Directional-derivative vector is tangent to manifold in direction.
Function derivatives {rate, differentiation} can be with respect to time.
Curve gradient can be indeterminate at point {singular point, curve}.
Relative minima {minimum, curve} have first derivative zero and second derivative greater than zero.
Relative maxima {maximum, curve} have first derivative zero and second derivative less than zero.
Functions {concave curve} can have points where second derivative is greater than zero {concave upward, concave curve} {convex downward, concave curve} and tangent is below curve. Functions can have points where second derivative is less than zero {concave downward, concave curve} {convex upward, concave curve} and tangent is above curve.
Functions {convex curve} can have points where second derivative is greater than zero {concave upward, convex curve} {convex downward, convex curve} and tangent is below curve. Functions can have points where second derivative is less than zero {concave downward, convex curve} {convex upward, convex curve} and tangent is above curve.
Functions can have points {optimum, calculus} {optima} where they are greatest or least. Functions can have highest points {relative maximum} in intervals. Functions can have lowest points {relative minimum} in intervals.
derivative
At maximum or minimum, derivative equals zero or has no definition. If first derivative changes sign, point is relative maximum or minimum.
Two-variable function maximum and minima are at points where both partial derivatives are zero. Maximum is if second partial derivative with respect to variable is less than zero: D^2f(x,y) / Dx < 0, where D^2 is second partial derivative, D is partial derivative, x and y are variables, and f is function. Minimum is if second partial derivative with respect to variable is greater than zero: D^2f(x,y) / Dx > 0.
Product of second partial derivatives with respect to each variable minus product of second derivatives with respect to each variable must be greater than zero: (D^2f(x,y) / Dx) * (D^2f(x,y) / Dy) - (d^2f(x,y) / dx^2) * (d^2f(x,y) / dy^2) > 0, where D^2 is second partial derivative, D is partial derivative, d^2 is second derivative, d is derivative, x and y are variables, and f is function.
Function can have points {inflection point}| where second derivative equals zero, tangent intersects curve, and curve has zero curvature. At inflection point, curve changes from concave to convex, or vice versa.
If a continuous function has derivatives at all interval points, at one or more points derivative has slope equal to slope of straight line passing through interval endpoints {mean value theorem}. For interval (a,b), line slope is (f(b) - f(a)) / (b - a). Point (x,y) has a <= x <= b and dy/dx = (f(b) - f(a)) / (b - a).
For two functions over interval, at one or more points derivative ratio equals difference ratio {extended law of the mean} {law of the mean extended}. For interval (a,b), and functions f(x) and g(x), at (x,y), (f(b) - f(a)) / (g(b) - g(a)) = f'(x) / g'(x).
If continuous function has two roots over interval and has derivatives at all interval points, first derivative equals zero at one or more points {Rolle's theorem} {Rolle theorem}.
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.
For constants {constant, differentiation}, function does not change, so derivative is zero. f(x) = c, and df(x) / dx = 0.
For constant times function {constant times function differentiation}, derivative is constant times function derivative. y = c * f(x), so dy / dx = c * df(x) / dx.
For power functions {power function differentiation}, reduce exponent by one and multiply by original exponent: dx^n = n * x^(n - 1) * dx. For example y = x^3, dy / dx = 3 * x^2.
d(e^x) / dx = e^x {exponential function, differentiation}. d(e^u(x)) = e^u(x) * du(x). Limit of (1 + 1/n)^n is e. Limit of (1 + h)^(1/h) is e. Limit of (1 + dx)^(1/dx) is e. Limit of (1 + (x / dx))^(x / dx) is e. On semi-log graph paper, y = b * a^x makes straight lines. On log-log graph paper, y = b * x^a makes straight lines.
If x > 0, dln(x) = 1/x {logarithmic function, differentiation}. If u(x) > 0, dln(u(x)) = (1 / u(x)) * du(x). (v(x))^a = e^(a * ln(v(x))). (v(x))^z(x) = e^(z(x) * ln(v(x))).
dsin(x) = cos(x) {trigonometric function, differentiation}. dcos(x) = -sin(x). dtan(x) = (sec(x))^2. dcot(x) = -(csc(x))^2. dsec(x) = sec(x) * tan(x). dcsc(x) = -csc(x) * cot(x).
darcsin(x) = 1 / (1 - x^2)^0.5 {trigonometric function, inverse differentiation}. darccos(x) = -1 / (1 - x^2)^0.5. darctan(x) = 1 / (1 + x^2). darccot(x) = -1 / (1 + x^2). darcsec(x) = 1 / (x * (x^2 - 1)^0.5). darccsc(x) = -1 / (x * (x^2 - 1)^0.5).
sinh(x) = (e^x - e^-x) / 2 and cosh(x) = (e^x + e^-x) / 2 {hyperbolic function, differentiation}. dsinh(x) = cosh(x). dcosh(x) = sinh(x).
For sum of terms {sum of terms differentiation}, find sum of differentials. For h(x) = f(x) + g(x), dh(x) = df(x) + dg(x).
For product of functions {product of functions differentiation}, add second function times first-function differential and first function times second-function differential: g(x) * df(x) + f(x) * dg(x).
For quotient of functions {quotient of functions differentiation}, multiply second function by first-function differential: g(x) * df(x). Then subtract first function times second-function differential: g(x) * df(x) - f(x) * dg(x). Then divide by second function squared: (g(x) * df(x) - f(x) * dg(x)) / (g(x))^2.
Vector functions {gradient, vector}| can be sum of each partial derivative times its unit vector i: (Df(x,y) / Dx) * i + (Df(x,y) / Dy) * j, where D is partial derivative. Gradient is in respect to direction. Gradient uses an operator {del operator}, which is upside-down uppercase delta: del = ((D / Dx) * i + (D / Dy) * j). For two dimensions, gradient is normal vector to vector-function curve. For three dimensions, gradient is normal vector to vector-function surface.
Vector functions {curl, vector}| can be vector products of del operator and vector function: del x f. Curl of gradient of scalar function equals zero: del x (del f) = 0.
Scalar functions {divergence, vector}| can be scalar products of del operator and vector function: del . f. Divergence of curl equals zero: del . (del x f) = 0.
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