A function f:E→R defined on a set E⊂R is differentiable at a point x ∈ E that is a limit point of E if f(x+h)−f(x)=A(x)h+α(x;h), where h↦A(x)h is a linear function in h and α(x;h)=o(h)as h→0, x+h∈E.
Definition 2
The function h↦A(x)h of Definition 1, which is linear in h, is called the differential of the functionf:E→R at the point x∈E and is denoted df(x) or Df(x). Thus, df(x)(h)=A(x)h.
We obtain
dx(h)df(x)(h)=f′(x)
We denote the set of all such vectors by TR(x0) or TRx0. Similarly, we denote by TR(x0) or TRy0 the set of all displacement vectors from the point y0 along the y-axis. It can then be seen from the definition of the differential that the mapping
df(x0):TR(x0)→TR(f(x0))
The derivative of an inverse function
If a function f is differentiable at a point x0 and its differential df(x0):TR(x0)→TR(y0)a is invertible at that point, then the differential of the function f−1 inverse to f exists at the point y0=f(x0) and is the mapping
df−1(y0)=[df(x0)]−1:TR(y0)→TR(x0)
inverse to df(x0):TR(x0)→TR(y0)a .
The derivative of some common function formula
(C)′=0
(xμ)′=μxμ−1
(sinx)′=cosx
(cosx)′=−sinx
(tanx)′=sec2x
(cotx)′=−csc2x
(secx)′=secxtanx
(cscx)′=−cscxcotx
(ax)′=axlna(a>0,a=1)
(ex)′=ex
(logax)′=xlna1(a>0,a=1)
(lnx)′=x1
(arcsinx)′=1−x21
(arccosx)′=−1−x21
(arctanx)′=1+x21
(arccotx)′=−1+x21
(sinhx)′=coshx
(coshx)′=sinhx
(tanhx)′=cosh2x1
(cothx)′=−sinh2x1
(arsinhx)′=(ln(x+1+x2))′=1+x21
(arcoshx)′=(ln(x±x2−1))′=±x2−11
(artanhx)′=(21ln1−x1+x)′=1−x21
(arcothx)′=(21lnx−1x+1)′=x2−11
theorem Leibniz
L'Hôpital's rule
The theorem states that for functions f and g which are differentiable on an open interval I except possibly at a point c contained in I, if limx→cf(x)=limx→cg(x)=0 or ±∞,limx→cf(x)=limx→cg(x)=0or ±∞ ,and g′(x)=0g′(x)=0 for all x in I with x=c, and limx→cg′(x)f′(x) exists, then
Suppose f,gare the function that defined in the set X, and also have the nth derivative, s.t.
(fg)(n)(x)=k=0∑n(nk)f(k)(x)g(n−k)(x)
x→climg(x)f(x)=x→climg′(x)f′(x)
mean value theorem
Taylor's theorem
Rolle's theorem
Let k≥1 be an integer and let the function f:R→R be k times differentiable at the point a∈R. Then there exists a function Rk:R→R such that ,
If a real-valued function f is continuous on a closed interval [a,b], differentiable on the open interval (a,b), and f(a)=f(b), then there exists at least one c in the open interval (a,b) such that f′(c)=0.
If a real-valued function f is continuous on a closed interval [a,b], differentiable on the open interval (a,b), then there is exists at least one c in the open interval (a,b) such that
f(b)−f(a)=f′(c)(b−a)
Cauchy’s Mean Value Theorem
If a real-valued function f is continuous on a closed interval [a,b], differentiable on the open interval (a,b), and g′(x)=0 for all x∈(a,b) , then there is exists at least one c in the open interval (a,b) such that