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x7→ √x2−x3x7→ (x2−1) arccos(x2)
R
x7→ x|x|x7→ x
|x|+1
R
f:x7→ (xsin(1/x)x6= 0
0g:x7→ (x2sin(1/x)x6= 0 0
x7→ arctan x
x2+1 x7→ 1 (x+1)2x7→ sin x (cos x+2)4
x7→ xxx7→ (ch x)xx7→ ln |x|
f1(x) = arctan (ex), f2(x) = arctan (sh x)f3(x) = arctan th x
2
f: [0 ; π/2] →R
f(x) = √sin x+x
f f−1
f:R→R
∀(x, y)∈R2, f(x+y) = f(x) + f(y)
f I a I
1
2h(f(a+h)−f(a−h))
h
f a
f a
a

n
x7→ x2(1 + x)nx7→ (x2+ 1)ex
n
x7→ 1
1−x, x 7→ 1 1 + xx7→ 1 1−x2
n
x7→ 1
1−x2
n x 7→ cos3x
n t 7→ cos(t)et
f:R→Rf(x)=ex√3sin x
f(n)(x)=2nex√3sin x+nπ
6
n xn−1e1/x
(−1)nx−(n+1)e1/x
f:x7→ arctan x
n≥1
f(n)(x)=(n−1)! cosn(f(x)) sin(nf(x) + nπ/2)
f(n)n≥1
n x 7→ x2n
n
X
k=0 n
k2
f:R→Rf0
f
a, b, c ∈Rx∈]0 ; 1[
4ax3+ 3bx2+ 2cx =a+b+c
f: [a;b]→Rf0(a)>0f0(b)<0
f
n∈Nf:I→RCnn+ 1
I
n f I
α(n−1) f0+αf
I

f:x7→ (x2−1)n(n)
f n
f(1) f(−1)
f n ]−1 ; 1[
f:I→RI a, b, c
I
∃d∈I, f(a)
(a−b)(a−c)+f(b) (b−c)(b−a)+f(c) (c−a)(c−b)=1 2f00(d)
n∈Na < b ∈Rf: [a;b]→Rn
f(a) = f0(a) = . . . =f(n−1)(a)=0 f(b) = 0
c∈]a;b[f(n)(c) = 0
f:R→R
lim
−∞ f= lim +∞f= +∞
c∈Rf0(c)=0
f: [0 ; +∞[→R
lim
+∞f=f(0)
c > 0f0(c)=0
a > 0f[0 ; a] ]0 ; a]
f(0) = 0 f(a)f0(a)<0
c∈]0 ; a[f0(c) = 0
a > 0f: [0 ; a]→R
f(0) = f(a)=0 f0(0) = 0
x7→ f(x)/x ]0 ; a[
f
f: [a;b]→R
f(a) = f(b) = 0 f0(a)>0, f0(b)>0
c1, c2, c3∈]a;b[c1< c2< c3
f0(c1) = f(c2) = f0(c3) = 0
f: [a;b]→RC2
f(a) = f0(a)f(b) = f0(b)
c∈]a;b[
f(c) = f00(c)
f(x)f0(x) ex

f:I→R
f
fC2[a;a+ 2h]a∈Rh > 0
∃c∈]a;a+ 2h[, f(a+ 2h)−2f(a+h) + f(a) = h2f00(c)
ϕ(x) = f(x+h)−f(x)
lim
x→+∞(x+ 1)e 1
x+1 −xe1
x
n+1
√n+ 1 −n
√n∼ −ln n
n2
∀x > 0,1
1 + x<ln(1 + x)−ln(x)<1 x
k∈N\ {0,1}
lim
n→∞
kn
X
p=n+1
1
p
f∈ C2(R+,R) limx→+∞f(x) = a∈R
f00 f0(x)x→+∞
∀x∈]−1 ; +∞[,x
1+x≤ln(1 + x)≤x
∀x∈R+,ex≥1 + x+x2
2
p∈]0 ; 1]
t≥0
(1 + t)p≤1 + tp
x, y ≥0
(x+y)p≤xp+yp
f: [a;b]→RC1f
f:R→CC1
f
f:R+→R
f(x) = (x2ln x x 6= 0
0x= 0
C1R+

n∈N
fn:x7→ (xn+1 x≥0
0
CnR
f:R+→RC2f0(0) = 0
g:R+→RC1
∀x∈R+, f(x) = g(x2)