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176
Основные правила дифференцирования функций
Функция
)(
x
f
y =
Производная
x
y
y
d
d
=
′
Функция
)(
x
f
y
=
Производная
x
y
y
d
d
=
′
u ± v u ′ ± v′ C
⋅
u C⋅u′
u
⋅
v
u
⋅
v ′+v
⋅
u′
v
u
2
v
vuuv
′
−
′
u
1
2
u
u
′
−
)(u
f
u
du
udf
⋅
)(
n
u uun
n
′
⋅
⋅
−1
u
u
u
2
′
v
u
vuuuuv
vv
⋅⋅+
′
⋅⋅
−
)ln(
1
)ln(u
u
u
′
)sin(u
uu
′
⋅
)cos(
)cos(u
uu
′
⋅− )sin(
)(sin
2
u
uu
′
⋅
)2sin(
)(cos
2
u
uu
′
⋅− )2sin(
Интегралы функций
Cxdx +=
∫
; C
a
x
dxx
a
a
+
+
=
+
∫
1
1
(
)
1
−
≠
a ;
Cx
x
dx
dxx +==
∫∫
−
ln
1
;
()
Cbax
abax
dx
++=
+
∫
ln
1
;
() ()
Ckx
k
dxkx +−=
∫
cos
1
sin ;
() ()
Ckx
k
dxkx +=
∫
sin
1
cos .
Основные правила дифференцирования функций
Функция dy Функция Производная
y = f ( x) Производная y ′ = y = f ( x)
dx dy
y′ =
dx
u ± v u ′ ± v′ C⋅u C⋅u′
u vu ′ − uv ′
u⋅ v u⋅v ′+v⋅u′
v v2
1 u′ df (u )
− f (u ) ⋅u
u u2 du
u′
un n ⋅ u n −1 ⋅ u′ u
2 u
u′
uv v ⋅ u v −1 ⋅ u′ + u v ⋅ ln(u ) ⋅ v ln(u )
u
sin(u ) cos(u ) ⋅ u′ cos(u ) − sin(u ) ⋅ u′
sin 2 (u ) sin(2u ) ⋅ u′ cos2 (u ) − sin(2u ) ⋅ u′
Интегралы функций
x a +1
∫ dx = x + C ; ∫ x a dx = + C (a ≠ −1) ;
a +1
dx dx 1
∫ x −1dx = ∫ = ln x + C ; ∫ = ln(ax + b ) + C ;
x ax + b a
1 1
∫ sin(kx)dx = − k cos(kx) + C ; ∫ cos(kx)dx = k sin(kx) + C .
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