Richiami di teoria: derivate Derivate delle funzioni elementari f (x) = k , f (x) = xα ,
f ′ (x) = 0
k∈R
f ′ (x) = αxα−1
α∈R
f ′ (x) = ex
f (x) = ex f (x) = ax ,
f ′ (x) = ax log a
a>0
f (x) = log x f (x) = loga x ,
a > 0 , a 6= 1
f ′ (x) =
1 x
f ′ (x) =
1 x log a
f (x) = sin x
f ′ (x) = cos x
f (x) = cos x
f ′ (x) = − sin x
f (x) = arctan x
f ′ (x) =
f (x) = arccotx
f ′ (x) = −
f (x) = arcsin x
f ′ (x) = √
f (x) = arccos x
f ′ (x) = − √
1 1 + x2 1 1 + x2 1 1 − x2 1 1 − x2
f (x) = sinh x =
ex − e−x 2
f ′ (x) = cosh x
f (x) = cosh x =
ex + e−x 2
f ′ (x) = sinh x 1
2
RICHIAMI DI TEORIA: DERIVATE
Regole di derivazione (f + g)′ (x) = f ′ (x) + g′ (x) (f g)′ (x) = f ′ (x)g(x) + f (x)g′ (x) (f ◦ g)′ (x) = f ′ (g(x)) g′ (x) ′
f g
(x) =
′
f ′ (x)g(x) − f (x)g′ (x) [g(x)]2
f −1 (x) =
1 f ′ (f −1 (x))