Troisième année niveau 2 Dérivée 2 Série 6

1) f(x) = x4+ 3x2 −6f0(x) = 4x3+ 6x
2) f(x)=6x3 −x2f0(x) = 18x2 −2x
3) f(x) = x5
a+b−
x2
a−bf0(x) = 5x4 a+b−
2x
a−b
4) f(x) = x3
−x2+ 1
5f0(x) = 3x2
−2x
5
5) f(x)=2ax3 −
x2
b+c f0(x) = 6ax2
−
2x
b
6) f(x)=6x7/2+ 4x5/2+ 2x f0(x) = 21x5/2+ 10x3/2+ 2
7) f(x) = √3x+3
√x+1
xf0(x) = √3
2√x+1
33
√x2−
1
x2
8) f(x) = (x+ 1)3
x3/2f0(x) = 3(x+ 1)2(x−1) 2x5/2
9) f(x) = x
m+m x+x2 n2+n2 x2f0(x) = 1 m−
m
x2+2x n2−
2n2
x3
10) f(x) = 3
√x2−2√x+ 5 f0(x) = 2
33
√x−
1
√x
11) f(x) = ax2
3
√x+b
x√x−
3
√x
√xf0(x) = 5
3ax2/3
−
3
2bx−5/2+1 6x−7/6
12) f(x) = (1 + 4x3)(1 + 2x2)f0(x) = 4x(1 + 3x+ 10x3)
13) f(x) = x(2x−1)(3x+ 2) f0(x) = 2(9x2+x−1)
14) f(x) = (2x−1)(x2 −6x+ 3) f0(x) = 6x2 −26x+ 12
15) f(x) = 2x4
b2−x2f0(x) = 4x3(2b2
−x2)
(b2−x2)2
16) f(x) = a−x
a+xf0(x) = −
2a
(a+x)2
17) f(t) = t3
1 + t2f0(t) = t2(3 + t2) (1 + t2)2
18) f(s) = (s+ 4)2
s+ 3 f0(s) = (s+ 2)(s+ 4) (s+ 3)2

19) f(x) = x2+ 2
x2−x−2f0(x) = x4
−2x3 −6x2 −4x+ 2
(x2−x−2)2
20) f(x) = xp
xm−amf0(x) = xp−1((p−m)xm−pam) (xm−am)2
21) f(x) = (2x2 −3)2f0(x) = 8x(2x2 −3)
22) f(x)=(x2+a2)5f0(x) = 10x(x2+a2)4
23) f(x) = px2+a2f0(x) = x
√x2+a2
24) f(x)=(a+x)√a−x f0(x) = a−3x
2√a−x
25) f(x) = r1 + x
1−xf0(x) = −
1
(1 −x)√1−x2
26) f(x) = 2x2
−1
x√1 + x2f0(x) = 1+4x2
x2p(1 + x2)3
27) f(x) = 3
px2+x+ 1 f0(x) = 2x+ 1
33
p(x2+x+ 1)2
28) f(x) = (1 + 3
√x)3f0(x) = 1 + 1
3
√x2
29) f(x) = rx+qx+√x f0(x) = 1
2qx+px+√x 1 + 1
2px+√x1 + 1
2√x!
30) f(x) = sin2x f0(x) = sin 2x
31) f(x) = 2 sin x+ cos 3x f0(x) = 2 cos x−3 sin 3x
32) f(x) = tan(ax +b)f0(x) = a
cos2(ax +b)
33) f(x) = sin x
1 + cos xf0(x) = 1 1 + cos x
34) f(x) = sin 2xcos 3x f0(x) = 2 cos 2xcos 3x−3 sin 2xsin 3x
35) f(x) = cot25x f0(x) = −10 cot 5xcsc25x
36) f(t) = tsin t+ cos t f0(t) = tcos t
37) f(x) = sin3tcos t f0(x) = sin2t(3 cos2t−sin2t)

38) f(φ) = asin3φ
3f0(φ) = asin2φ 3cos φ 3
39) f(x) = a√cos 2x f0(x) = −
asin 2x
√cos 2x
40) f(x) = tan x
2+ cot x 2
xf0(x) = −
2xcos x+ sin2xtan x
2+ cot x 2
x2sin2x
41) f(x) = a1−cos2x
22
f0(x) = 2asin3x
2cos x 2
42) f(x) = 1
2tan2x f0(x) = tan xsec2x
43) f(x) = ln cos x f0(x) = −tan x
44) f(x) = ln tan x f0(x) = 2
sin 2x
45) f(x) = ln sin2x f0(x) = 2 cot x
46) f(x) = tan x−1
sec xf0(x) = sin x+ cos x
47) f(x) = ln r1 + sin x
1−sin xf0(x) = 1 cos x
48) f(x) = sin ln x f0(x) = cos ln x
x
49) f(x) = sin cos x f0(x) = −sin xcos(cos x)
50) f(x) = ln(ax +b)f0(x) = a
ax +b
51) f(x) = ln 1 + x
1−xf0(x) = 2 1−x2
52) f(x) = ln(x2 −sin x)f0(x) = 2x−cos x
x2−sin x
53) f(x) = ln(x3 −2x+ 5) f0(x) = 3x2
−2
x3−2x+ 5
54) f(x) = xln(x)f0(x) = ln x+ 1
55) f(x) = ln3x f0(x) = 3 ln x
x
56) f(x) = ln(x+p1 + x2)f0(x) = 1
√1 + x2
57) f(x) = ln ln x f0(x) = 1
xln x

58) f(x) = ln r1 + x
1−xf0(x) = 1 1−x2
59) f(x) = sin x
2 cos2xf0(x) = 1 + sin2x 2 cos3x
60) f(x) = eax f0(x) = aeax
61) f(x) = e4x+5 f0(x) = 4e4x+5
62) f(x) = ae√xf0(x) = a
2√xe√x
63) f(x) = ex(1 −x2)f0(x) = ex(1 −2x−x2)
64) f(x) = ex−1
ex+ 1 f0(x) = 2ex (ex+ 1)2
65) f(x) = ln ex
1 + exf0(x) = 1 1 + ex
66) f(x) = a
2(e
x
a−e−x a)f0(x) = 1
2(e
x
a−e−x a)
67) f(x) = esin xf0(x) = esin xcos x
68) f(x) = ecos xsin x f0(x) = ecos x(cos x−sin2x)
69) f(x) = xnesin xf0(x) = xn−1esin x(n+xcos x)
70) f(x) = arcsin x
af0(x) = 1
√a2−x2
71) f(x) = (arcsin x)2f0(x) = 2 arcsin x
√1−x2
72) f(x) = arctan 2x
1−x2f0(x) = 2 1 + x2
73) f(x) = arccos(x2)f0(x) = −2x
√1−x4
74) f(x) = pa2−x2+aarcsin x
af0(x) = ra−x a+x
75) f(x) = xarcsin x f0(x) = arcsin x+x
√1−x2