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.
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Date Modified: 2022.0225