1.1 INTRODUCTION ET RÉVISION

(1)

1.1 INTRODUCTION

ET RÉVISION

(2)

Aujourd’hui, nous allons voir

(3)

Aujourd’hui, nous allons voir

Survol des cours calcul différentiel, calcul intégral et

(4)

Aujourd’hui, nous allons voir

Survol des cours calcul différentiel, calcul intégral et algèbre linéaire.

(5)

y x

= f (a + x) f (a) x

(6)

y x

= f (a + x) f (a) x

(7)

y x

= f (a + x) f (a) x

(8)

y x

= f (a + x) f (a) x

(9)

y x

= f (a + x) f (a) x

(10)

(11)

(12)

(13)

Les règles de dérivations

(14)

(15)

(16)

(17)

(18)

(19)

d

dx (kf (x)) = k d

dx f (x)

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(20)

d

dx (kf (x)) = k d

dx f (x)

d

dx (f (x) + g(x)) = df (x)

dx + dg(x) dx

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(21)

d

dx (kf (x)) = k d

dx f (x)

d

dx (f (x) + g(x)) = df (x)

dx + dg(x) dx

d (f (x)g(x)) = df (x)

g(x) + f (x) dg(x)

(22)

d

dx (kf (x)) = k d

dx f (x)

d

dx (f (x) + g(x)) = df (x)

dx + dg(x) dx

d

dx (f (x)g(x)) = df (x)

dx g(x) + f (x) dg(x) dx

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✓ ◆ ^df^(x)

g(x) f (x) ^dg^(x)

(23)

d

dx (kf (x)) = k d

dx f (x)

d

dx (f (x) + g(x)) = df (x)

dx + dg(x) dx

d

dx (f (x)g(x)) = df (x)

d dx

✓ f (x) g(x)

◆

=

df(x)

dx g(x) f (x) ^dg_dx^(x) g²(x)

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(24)

d

dx (kf (x)) = k d

dx f (x)

d

dx (f (x) + g(x)) = df (x)

dx + dg(x) dx

d

dx (f (x)g(x)) = df (x)

d dx

✓ f (x) g(x)

◆

=

df(x)

dx g(x) f (x) ^dg_dx^(x) g²(x)

<latexit sha1_base64="ynhhs1ISNpg7zkQqz2+80yQ85nc=">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</latexit><latexit sha1_base64="ynhhs1ISNpg7zkQqz2+80yQ85nc=">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</latexit><latexit sha1_base64="ynhhs1ISNpg7zkQqz2+80yQ85nc=">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</latexit><latexit sha1_base64="ynhhs1ISNpg7zkQqz2+80yQ85nc=">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</latexit>

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Exemple

^f ^{(x) =} ³^x ^sin(x²⁾

ln x

(46)

Exemple

^f ^{(x) =} ³^x ^sin(x²⁾

ln x

f ⁰(x) = (3^x sin(x²))⁰ ln x (3^x sin(x²)) ln x⁰ ln² x

(47)

Exemple

^f ^{(x) =} ³^x ^sin(x²⁾

ln x

(48)

Exemple

^f ^{(x) =} ³^x ^sin(x²⁾

ln x

(3^x⁰ sin(x²) + 3^x sin(x²)⁰) ln x (3^x sin(x²)) ¹ f ⁰(x) = (3^x sin(x²))⁰ ln x (3^x sin(x²)) ln x⁰

ln² x

(49)

Exemple

^f ^{(x) =} ³^x ^sin(x²⁾

ln x

ln² x

(50)

Exemple

^f ^{(x) =} ³^x ^sin(x²⁾

ln x

ln² x

(51)

Exemple

^f ^{(x) =} ³^x ^sin(x²⁾

ln x

(3^x ln 3 sin(x²) + 3^x cos(x²)(x²)⁰) ln x (3^x sin(x²)) ¹

= (3^x⁰ sin(x²) + 3^x sin(x²)⁰) ln x (3^x sin(x²)) _x¹ ln² x

(52)

Exemple

^f ^{(x) =} ³^x ^sin(x²⁾

ln x

(53)

Exemple

^f ^{(x) =} ³^x ^sin(x²⁾

ln x

(54)

Exemple

^f ^{(x) =} ³^x ^sin(x²⁾

ln x

= (3^x ln 3 sin(x²) + 3^x cos(x²)(x²)⁰) ln x (3^x sin(x²)) _x¹ ln² x

(3^x ln 3 sin(x²) + 3^x cos(x²)2x) ln x (3^x sin(x²)) ¹

(55)

Exemple

^f ^{(x) =} ³^x ^sin(x²⁾

ln x

= (3^x ln 3 sin(x²) + 3^x cos(x²)(x²)⁰) ln x (3^x sin(x²)) _x¹ ln² x

(3^x ln 3 sin(x²) + 3^x cos(x²)2x) ln x (3^x sin(x²)) ¹

(56)

Faites les exercices suivants

Calculer le dérivée des fonctions suivantes:

1)

2)

3)

f<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> (x) = 4 tan³(xe^x)

f (x) = log₅(p

4x⁷ 3) sin x

f (x) = 4^x p⁵

x³ sin x + csc³ x log₆ ⇡²

4^cos(x²⁺¹⁾ !

(57)

différentielle

(58)

différentielle

(59)

différentielle Lorsque <latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> x ⇡ 0

(60)

y

x ⇡ dy dx

(61)

y

x ⇡ dy dx

(62)

différentielle

dy

dx = f ⁰(x)

Lorsque <latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> x ⇡ 0

y

x ⇡ dy dx

(63)

différentielle

dy

dx = f ⁰(x)

dy = f ⁰(x)dx

Lorsque <latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> x ⇡ 0

y

x ⇡ dy dx

(64)

Intégrale indéfinie Z

f (x)dx = F (x) + C

(65)

f (x)dx = F (x) + C

où ^F<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> ⁰^{(x) =} ^f ^(x)

(66)

Z

f ⁰(x)dx = f (x) + C

f (x)dx = F (x) + C

(67)

Z

f ⁰(x)dx = f (x) + C

f (x)dx = F (x) + C