Formulaire de Dérivation

Récapitulatif des formules de dérivées

Conventions: u = u(x) et v = v(x) sont des fonctions dérivables, a, c, n, α des constantes, et f′ = df/dx

1. Règles générales

(c)′ = 0
(a u)′ = a u′
(u + v)′ = u′ + v′
(u - v)′ = u′ - v′
(u v)′ = u′v + uv′
(u / v)′ = (u′v - uv′) / v²
(1 / v)′ = -v′ / v²
(u v w)′ = u′vw + uv′w + uvw′

2. Fonctions usuelles

(x)′ = 1
(xⁿ)′ = n x^n-1
(x^α)′ = α x^α-1
(1 / x)′ = -1 / x²

3. Exponentielles et logarithmes

(e^x)′ = e^x
(a^x)′ = a^x ln a
(ln x)′ = 1 / x
(ln|x|)′ = 1 / x

4. Fonctions trigonométriques et autres propriétés

(sin x)′ = cos x
(cos x)′ = -sin x
(tan x)′ = 1 + tan²x = 1 / cos²x
(uⁿ)′ = n u^n-1u′
(f ˆ u)′ = (f′ ˆ u) · u′
dy/dx = (dy/du) · (du/dx)
(f^-1)′(x) = 1 / f′(f^-1(x))
(uv)′ / (uv) = u′/u + v′/v
(u^v)′ = u^v(v′ ln u + v(u′/u))
(uv)⁽ⁿ⁾ = Σ_k=0ⁿ C(n,k) u^(k)v^(n-k)

(1 / xⁿ)′ = -n / xⁿ⁺¹
(√x)′ = 1 / (2√x)
(ⁿ√x)′ = (1/n) x^(1/n)-1
(|x|)′ = x / |x| (x ≠ 0)
(log_ax)′ = 1 / (x ln a)
(e^u)′ = u′e^u
(a^u)′ = u′a^u ln a
(ln u)′ = u′ / u
(log_au)′ = u′ / (u ln a)
(cot x)′ = -1 - cot²x = -1 / sin²x
(sec x)′ = sec x tan x
(csc x)′ = -csc x cot x

5. Fonctions trigonométriques réciproques

arccot′(x) = -1 / (1 + x²)
(arcsin x)′ = 1 / √(1 - x²)
(arccos x)′ = -1 / √(1 - x²)
(arctan x)′ = 1 / (1 + x²)

6. Fonctions hyperboliques

(sinh x)′ = cosh x
(cosh x)′ = sinh x
(tanh x)′ = 1 - tanh²x = 1 / cosh²x

7. Fonctions hyperboliques réciproques

arcsec′(x) = 1 / (|x|√(x² - 1))
arccsc′(x) = -1 / (|x|√(x² - 1))
(coth x)′ = 1 - coth²x = -1 / sinh²x
sech′(x) = -sech x tanh x
csch′(x) = -csch x coth x
(argcoth x)′ = 1 / (1 - x²)
(arg sinh x)′ = 1 / √(x² + 1)
(argsech x)′ = -1 / (x√(1 - x²))
(argcosh x)′ = 1 / √(x^2 - 1)
(argcsch x)′ = -1 / (|x|√(1 + x²))
(argtanh x)′ = 1 / (1 - x²)

8. Formes composées (avec u = u(x))

(tan u)′ = u′ / cos²u
(uⁿ)′ = n u^n-1u′
(cot u)′ = -u′ / sin²u
(√u)′ = u′ / (2√u)
(arcsin u)′ = u′ / √(1 - u²)
(1 / u)′ = -u′ / u²
(e^u)′ = u′e^u
(arccos u)′ = -u′ / √(1 - u²)

(ln u)′ = u′ / u
(arctan u)′ = u′ / (1 + u²)
(sinh u)′ = u′cosh u
(a^u)′ = u′a^u ln a
(cosh u)′ = u′sinh u
(sin u)′ = u′cos u
(cos u)′ = -u′sin u
(tanh u)′ = u′ / cosh²u