CHAPITRE 6 : DÉRIVATION

(d)
(d)y=ax +b,
a, b
x y M(x, y)
•a
A(xA;yA)B(xB;yB)
a=yB−yA
xB−xA
.
a= 0
•b
f:x7→ ax +b
y=ax +b

f I
Cf
a, b ∈I a 6=b
f a b
τa,b =f(b)−f(a)
b−a.
A B
A(a;f(a)) B(b, f(b))
f a b
(AB)
B
A
B A (AB)
a
a b a a +h
h0
h0
h6= 0
τa,a+h=f(a+h)−f(a)
h
l
f a f a
f0(a)l
f0(a) = lim
h→0
f(a+h)−f(a)
h.

f a f a
f a
f f(x) = x2−x+1 f(2)
f2 2 + h h = 2 h= 1 h= 0.5h= 0.1
f0(2) f

f:x7→ |x|(0; 0)
0
0
f:x7→ √x0
0h+∞h
0 0
0
f x0Tx0f
x0
Tx0:y=f0(x0)x+f(x0)−f0(x0)x0
y=f0(x0)(x−x0) + f(x0)
Tx0f0(x0)
y=f0(x0)x+b b x0
x0f(x0)
Tx0
f(x0) = f0(x0)x0+b⇐⇒ b=f(x0)−f0(x0)x0.
f0(x0)
f
f(0) f(1) f(3)
f0(0) f0(1) f0(3)
T0T1T3
f I f
I
x7→ f0(x)
f f0

f
f I f I x ∈I
f0(x)=0
f
x7→ x
f:x7→ xRx∈Rf0(x) = 1
y=x
1
x7→ x2
f:x7→ x2Rx∈Rf0(x)=2x
x∈Rx+h h ∈R
0
f(x+h)−f(x)
h=(x+h)2−x2 h
=x2+ 2xh +h2−x2
h
=h2+ 2xh
h
= 2x+h,
2x h 0
x7→ 1
x
f:x7→ 1
xR?x∈R?f0(x) = −1 x2
x∈Rx+h h ∈R
0
f(x+h)−f(x)
h=
1
x+h−1 x
h
=
x−(x+h)
x(x+h)
h
=−h
h(x2+xh)
=−1
x2+xh,
−1
x2h0