Themes > Science > Mathematics > Calculus > Derivatives > Differentiation 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)

(d/dx)c f(x) = c  Proof
(d/dx)f(x) (c is a constant)

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

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

(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 counter-clockwise from pos. x axis.


Information supplied by http://www.math2.org