Differentiation Identities

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Differentiation Identities

(Math | Calculus | Derivatives | Identities)

Definitions of the Derivative:
df / dx = lim (dx -> 0) (f(x+dx) - f(x)) / dx (right sided)
df / dx = lim (dx -> 0) (f(x) - f(x-dx)) / dx (left sided)
df / dx = lim (dx -> 0) (f(x+dx) - f(x-dx)) / (2dx) (both sided)

(d/dx) (integral) (a to x) f(t) dt = f(x) (Fundamental Theorem for Derivatives)

c f(x) = c

f(x) (c is a constant)

(d/dx) (f(x) + g(x)) = (d/dx) f(x) + (d/dx) g(x)

(d/dx) f(g(x)) = (d/dg) f(g) * (d/dx) g(x) (chain rule)

(d/dx) f(x)g(x) = f'(x)g(x) + f(x)g '(x) (product rule)

(d/dx) f(x)/g(x) = ( f '(x)g(x) - f(x)g '(x) ) / g^{^2}(x) (quotient rule)

Partial Differentiation Identities

if f( x(r,s), y(r,s) )

df / dr = df / dx * dx / DR + df / dy * dy / DR

df / ds = df / dx * dx / Ds + df / dy * dy / Ds

if f( x(r,s) )

df / DR = df / dx * dx / DR

df / Ds = df / dx * dx / Ds

directional derivative

df(x,y) / d(Xi sub a) = f1(x,y) cos(a) + f2(x,y) sin(a)

(Xi sub a) = angle counterclockwise from pos. x axis.