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1) Una funzione
y
=
f
(
x
)
{\displaystyle y=f(x)}
P
=
[
c
,
f
(
c
)
]
{\displaystyle \ P=[c,f(c)]}
lim
h
→
0
f
(
c
+
h
)
−
f
(
c
)
h
{\displaystyle \lim _{h\to 0}{f(c+h)-f(c) \over h}}
il quale limite si dice derivata di
f
(
x
)
{\displaystyle \ f(x)}
x
=
c
.
{\displaystyle \ x=c.}
Se il limite suddetto esiste per ogni valore
x
{\displaystyle \ x}
(
a
,
b
)
{\displaystyle \ (a,b)}
f
′
(
x
)
=
D
f
(
x
)
=
d
y
d
x
=
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
=
ϕ
(
x
)
,
{\displaystyle \ f'(x)=Df(x)={dy \over dx}=\lim _{h\to 0}{f(x+h)-f(x) \over h}=\phi (x),}
funzione che rappresenta la derivata di
f
(
x
)
{\displaystyle \ f(x)}
(
a
,
b
)
.
{\displaystyle \ (a,b).}
'esempio' Se
y
=
s
e
n
x
,
{\displaystyle \ y=\ sen\ x,}
f
(
x
+
h
)
−
f
(
x
)
h
=
s
e
n
(
x
+
h
)
−
s
e
n
(
x
)
h
=
2
s
e
n
h
2
c
o
s
(
x
+
h
2
)
h
,
{\displaystyle {f(x+h)-f(x) \over h}={\ sen(x+h)-\ sen(x) \over h}={2\ sen{h \over 2}\ cos(x+{h \over 2}) \over h},}
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
=
lim
h
→
0
[
s
e
n
h
2
h
2
c
o
s
(
x
+
h
2
)
]
=
c
o
s
x
{\displaystyle \lim _{h\to 0}{f(x+h)-f(x) \over h}=\lim _{h\to 0}[{\ sen{h \over 2} \over {h \over 2}}\ cos(x+{h \over 2})]=\ cos\ x}
onde
:
D
s
e
n
x
=
c
o
s
x
.
{\displaystyle :\qquad D\ sen\ x=\ cos\ x.}
2) Una funzione
y
=
f
(
x
)
{\displaystyle \ y=f(x)}
differenziabile
P
=
[
c
,
f
(
c
)
]
{\displaystyle \ P=[c,f(c)]}
Δ
y
{\displaystyle \ \Delta y}
P
{\displaystyle \ P}
Δ
y
=
A
h
+
ε
h
,
{\displaystyle \ \Delta y=Ah+\varepsilon h,}
essendo
A
{\displaystyle \ A}
ε
{\displaystyle \ \varepsilon }
h
.
{\displaystyle \ h.}
Se la precedente relazione vale in tutto
(
a
,
b
)
{\displaystyle \ (a,b)}
f
(
x
)
{\displaystyle \ f(x)}
(
a
,
b
)
.
{\displaystyle \ (a,b).}
d
y
=
A
h
=
f
′
(
x
)
h
=
f
′
(
x
)
d
x
{\displaystyle \ dy=Ah=f'(x)h=f'(x)dx}
d
y
d
x
=
f
′
(
x
)
{\displaystyle {dy \over dx}=f'(x)}
derivata di una somma o differenza
d
(
u
±
v
)
d
x
=
d
u
d
x
±
d
v
d
x
{\displaystyle {\frac {d(u\pm v)}{dx}}={\frac {du}{dx}}\pm {\frac {dv}{dx}}}
derivata di un prodotto
d
(
u
v
)
d
x
=
v
d
u
d
x
+
u
d
v
d
x
{\displaystyle {\frac {d(uv)}{dx}}=v\ {\frac {du}{dx}}+u\ {\frac {dv}{dx}}}
derivata di un quoziente
d
(
u
v
)
d
x
=
v
d
u
d
x
−
u
d
v
d
x
v
2
{\displaystyle {\frac {d({\frac {u}{v}})}{dx}}={\frac {v{\frac {du}{dx}}-u{\frac {dv}{dx}}}{v^{2}}}}
derivata di funzione di funzione
{\displaystyle }
derivata della funzione inversa di y=f(x)
d
x
d
y
=
1
d
y
d
x
,
s
e
d
y
d
x
≠
0
<
,
{\displaystyle {\frac {dx}{dy}}={\frac {1}{\frac {dy}{dx}}},\qquad se{\frac {dy}{dx}}\neq 0<,}
derivata di cf(x)
d
[
c
f
(
x
)
]
d
x
=
c
d
f
d
x
{\displaystyle {\frac {d[cf(x)]}{dx}}=c\ {\frac {df}{dx}}}
derivazione per serie Se una serie
∑
u
n
(
x
)
{\displaystyle \sum _{}^{}u_{n}(x)}
u
n
′
(
x
)
{\displaystyle u'_{n}(x)}
∑
u
n
′
(
x
)
{\displaystyle \sum _{}^{}u'_{n}(x)}
d
s
(
x
)
d
x
=
∑
i
u
n
i
′
(
x
)
,
{\displaystyle {ds(x) \over dx}=\sum _{i}^{}u'_{n_{i}}(x),}
essendo
s
(
x
)
{\displaystyle \ s(x)}
d
c
d
x
=
0
{\displaystyle {\frac {dc}{dx}}=0}
d
x
d
x
=
1
{\displaystyle {\frac {dx}{dx}}=1}
d
x
m
d
x
=
m
x
m
−
1
{\displaystyle {\frac {dx^{m}}{dx}}=mx^{m-1}}
d
[
f
(
x
)
]
m
d
x
=
m
[
f
(
x
)
]
m
−
1
d
f
d
x
{\displaystyle {\frac {d[f(x)]^{m}}{dx}}=m[f(x)]^{m-1}{\frac {df}{dx}}}
d
x
n
m
d
x
=
m
n
x
n
(
m
−
n
)
;
d
{\displaystyle {\frac {d{\sqrt[{n}]{x}}^{m}}{dx}}={\frac {m}{n}}{\sqrt[{n}]{x}}^{(m-n)};\qquad {\frac {d}{}}}
d
n
f
(
x
)
d
x
n
=
d
d
x
[
d
n
−
1
f
(
x
)
d
x
n
−
1
]
;
d
n
f
=
d
n
f
d
x
n
(
d
x
)
n
.
{\displaystyle {\frac {d^{n}f(x)}{dx^{n}}}={\frac {d}{dx}}[{\frac {d^{n-1}f(x)}{dx^{n-1}}}];\qquad d^{n}f={d^{n}f \over dx^{n}}(dx)^{n}.}
d
n
(
u
±
v
)
d
x
n
=
d
n
u
d
x
n
±
d
n
v
d
x
n
{\displaystyle {d^{n}(u\pm v) \over dx^{n}}={d^{n}u \over dx^{n}}\pm {d^{n}v \over dx^{n}}}
derivate parziali prime
δ
f
δ
x
=
lim
h
→
0
f
(
x
+
h
,
y
)
−
f
(
x
,
y
)
h
=
f
x
′
{\displaystyle {\frac {\delta f}{\delta x}}=\lim _{h\rightarrow 0}{\frac {f(x+h,y)-f(x,y)}{h}}=f'_{x}}
δ
f
δ
y
=
lim
k
→
0
f
(
x
,
y
+
k
)
−
f
(
x
,
y
)
k
=
f
y
′
{\displaystyle {\frac {\delta f}{\delta y}}=\lim _{k\rightarrow 0}\ {\frac {f(x,y+k)-f(x,y)}{k}}=f'_{y}}
derivate parziali seconde
∂
2
f
∂
x
2
=
∂
∂
x
(
∂
f
∂
x
)
;
∂
2
f
∂
y
2
=
∂
∂
y
(
∂
f
∂
y
)
{\displaystyle {\partial ^{2}f \over \partial x^{2}}={\partial \over \partial x}({\partial f \over \partial x});\qquad {\partial ^{2}f \over \partial y^{2}}={\partial \over \partial y}({\partial f \over \partial y})}
∂
2
f
∂
x
∂
y
=
∂
∂
y
(
∂
f
∂
x
)
;
∂
2
f
∂
y
∂
x
=
∂
∂
x
(
∂
f
∂
y
)
;
{\displaystyle {\partial ^{2}f \over \partial x\partial y}={\partial \over \partial y}({\partial f \over \partial x});\qquad {\partial ^{2}f \over \partial y\partial x}={\partial \over \partial x}({\partial f \over \partial y});}
∂
2
f
∂
x
∂
y
=
∂
2
f
∂
y
∂
x
,
{\displaystyle {\partial ^{2}f \over \partial x\partial y}={\partial ^{2}f \over \partial y\partial x},}
cioè le derivate seconde miste sono uguali. derivate di una funzione
z
=
f
(
x
,
y
)
{\displaystyle z=f(x,y)}
x
=
ϕ
(
t
)
,
y
=
ψ
(
t
)
{\displaystyle \ x=\phi (t),\ y=\psi (t)}
∂
f
∂
t
=
∂
f
∂
x
d
x
d
t
+
∂
f
∂
y
d
y
d
t
,
{\displaystyle {\partial f \over \partial t}={\partial f \over \partial x}{dx \over dt}+{\partial f \over \partial y}{dy \over dt},}
derivata secondo la direzione di coseni direttori
α
{\displaystyle \ \alpha }
β
{\displaystyle \ \beta }
α
,
β
,
γ
{\displaystyle \ \alpha ,\ \beta ,\ \gamma }
lim
Δ
r
→
0
Δ
f
Δ
r
=
d
f
d
r
=
∂
f
∂
x
α
+
∂
f
∂
y
β
;
lim
Δ
r
→
0
Δ
f
Δ
r
=
∂
f
∂
r
=
∂
f
∂
x
α
+
∂
f
∂
y
β
+
∂
f
∂
z
γ
,
{\displaystyle \lim _{\Delta r\to 0}{\Delta f \over \Delta r}={df \over dr}={\partial f \over \partial x}\alpha +{\partial f \over \partial y}\beta ;\qquad \lim _{\Delta r\to 0}{\Delta f \over \Delta r}={\partial f \over \partial r}={\partial f \over \partial x}\alpha +{\partial f \over \partial y}\beta +{\partial f \over \partial z}\gamma ,}
essendo
Δ
r
{\displaystyle \ \Delta r}
differenziali totali di
z
=
f
(
x
,
y
)
:
{\displaystyle \ z=f(x,y):}
d
f
=
∂
f
∂
x
d
x
+
∂
f
∂
x
d
y
;
d
n
f
=
(
∂
f
∂
x
d
x
+
∂
f
∂
y
d
y
)
(
n
)
,
{\displaystyle \ df={\partial f \over \partial x}dx+{\partial f \over \partial x}dy;\qquad d^{n}f=({\partial f \over \partial x}dx+{\partial f \over \partial y}dy)^{(n)},}
dove
(
n
)
{\displaystyle \ (n)}
ordine esponente
d
x
{\displaystyle \ dx}
d
y
.
{\displaystyle \ dy.}
Un'espressione:
A
(
x
,
y
)
d
x
+
B
(
x
,
y
)
d
y
{\displaystyle \ A(x,y)dx+B(x,y)dy}
f
(
x
,
y
)
{\displaystyle \ f(x,y)}
A
(
x
,
y
)
d
x
+
B
(
x
,
y
)
d
y
=
∂
f
∂
x
d
x
+
∂
f
∂
y
d
y
{\displaystyle \ A(x,y)dx+B(x,y)dy={\partial f \over \partial x}dx+{\partial f \over \partial y}dy}
per il che è necessario e sufficiente che sia verificata la condizione:
∂
A
∂
y
=
∂
B
∂
x
.
{\displaystyle {\partial A \over \partial y}={\partial B \over \partial x}.}
formula del valore medio per una funzione y=f(x) differenziabile nell'intervallo (x,x+h) :
f
(
x
+
h
)
−
f
(
x
)
=
h
f
′
(
x
+
θ
h
)
{\displaystyle \ f(x+h)-f(x)=hf'(x+\theta h)}
con
(
0
<
θ
<
1
)
{\displaystyle \ (\ 0<\theta \ <1)}
formula del valore medio per una funzione z=f(x,y) differenziabile in un campo C contenente il punto (x,y):
f
(
x
+
h
,
y
+
k
)
−
f
(
x
,
y
)
=
h
f
x
′
(
x
+
θ
h
,
y
+
θ
k
)
+
k
f
y
′
(
x
+
θ
h
,
y
+
θ
k
)
{\displaystyle \ f(x+h,y+k)-f(x,y)=hf'_{x}(x+\theta h,y+\theta k)+kf'_{y}(x+\theta h,y+\theta k)}
con
(
0
<
θ
<
1
)
{\displaystyle \ (0<\theta \ <1)}
formula di Cauchy per due funzioni f(x) e g(x) differenziabili nell'intervallo (x,x+h) e con g'(x)≠0
f
(
x
+
h
)
−
f
(
x
)
g
(
x
+
h
)
−
g
(
x
)
=
f
′
(
x
+
θ
h
)
g
′
(
x
+
θ
h
)
{\displaystyle {\frac {f(x+h)-f(x)}{g(x+h)-g(x)}}={\frac {f'(x+\theta h)}{g'(x+\theta h)}}}
con
(
0
<
θ
<
1
{\displaystyle \ (0<\theta \ <1}
formula di Taylor e di Mac-Laurin per una funzione f(y) differenziabile almeno fino all'ordine n :
f
(
x
+
h
)
=
f
(
x
)
+
h
1
f
′
(
x
)
+
h
2
2
!
f
″
(
x
)
+
.
.
.
+
h
n
n
!
f
(
n
)
(
x
+
θ
h
)
{\displaystyle \ f(x+h)=f(x)+{\frac {h}{1}}f'(x)+{\frac {h^{2}}{2!}}f''(x)+...+{\frac {h^{n}}{n!}}f^{(n)}(x+\theta h)}
f
(
x
)
=
f
(
0
)
+
x
1
!
f
′
(
0
)
+
x
2
2
!
f
″
(
0
)
+
.
.
.
+
x
n
n
!
f
(
n
)
(
θ
x
)
{\displaystyle \ f(x)=f(0)+{\frac {x}{1!}}f'(0)+{\frac {x^{2}}{2!}}f''(0)+...+{\frac {x^{n}}{n!}}f^{(n)}(\theta x)}
L'ultimo termine del secondo membro chiamasi termine complementare o resto. Si indica normalmente con
R
n
{\displaystyle \ R_{n}}
R
n
=
n
n
n
!
f
(
n
)
(
x
+
θ
h
)
(
f
o
r
m
a
d
i
L
a
g
r
a
n
g
e
)
{\displaystyle \ R_{n}={\frac {n^{n}}{n!}}f^{(n)}(x+\theta h)\qquad (forma\ di\ Lagrange)}
R
n
=
h
n
(
1
−
θ
)
(
n
−
p
)
f
(
n
)
(
x
+
θ
h
)
p
(
n
−
1
)
!
(
f
o
r
m
a
d
i
S
c
h
l
o
m
i
s
c
h
)
{\displaystyle \ R_{n}={\frac {h^{n}(1-\theta )^{(n-p)}f^{(n)}(x+\theta h)}{p(n-1)!}}\qquad (forma\ di\ Schlomisch)}
R
n
=
h
n
(
1
−
θ
)
n
−
1
f
(
n
)
(
x
+
θ
h
)
(
n
−
1
)
!
(
f
o
r
m
a
d
i
C
a
u
c
h
y
)
{\displaystyle \ R_{n}={\frac {h^{n}(1-\theta )^{n-1}f^{(n)}(x+\theta h)}{(n-1)!}}\qquad (forma\ di\ Cauchy)}
formule di Taylor e di Mac-Lorin per una funzione z=f(x,y) differenziabile fino all'ordine n :
regola di Hopital per le forme indeterminate:
Se il rapporto
f
(
x
)
ϕ
(
x
)
{\displaystyle {\frac {f(x)}{\phi (x)}}}
x=c si presenta nelle forme:
0
0
,
∞
∞
,
a
l
l
o
r
a
:
lim
x
→
c
f
(
x
)
ϕ
(
x
)
=
lim
x
→
c
f
′
(
x
)
ϕ
′
(
x
)
{\displaystyle {\frac {0}{0}},\quad {\frac {\infty }{\infty }},\quad allora:\quad \lim _{x\to \ c}{\frac {f(x)}{\phi (x)}}=\lim _{x\to \ c}{\frac {f'(x)}{\phi '(x)}}}
e se il limite del secondo membro esiste, e in caso contrario:
lim
x
→
c
f
(
x
)
ϕ
(
x
)
=
lim
x
→
c
f
(
n
)
(
x
)
ϕ
(
n
)
(
x
)
{\displaystyle \lim _{x\to \ c}{\frac {f(x)}{\phi (x)}}=\lim _{x\to \ c}{\frac {f^{(n)}(x)}{\phi ^{(n)}(x)}}}
essendo n il
I
n
s
e
r
i
s
c
i
q
u
i
u
n
a
f
o
r
m
u
l
a
{\displaystyle Inserisciquiunaformula}
Le forme indeterminate del tipo
0
⋅
∞
,
∞
−
∞
,
{\displaystyle 0\cdot \infty ,\infty -\infty ,}
a) mediante le trasformazioni:
f
(
x
)
⋅
ϕ
(
x
)
=
f
(
x
)
1
ϕ
(
x
)
f
(
x
)
−
ϕ
(
x
)
=
[
1
ϕ
(
x
)
−
1
f
(
x
)
]
:
1
f
(
x
)
ϕ
(
x
)
{\displaystyle f(x)\cdot \phi (x)={\frac {f(x)}{\frac {1}{\phi (x)}}}\qquad f(x)-\phi (x)=[{\frac {1}{\phi (x)}}-{\frac {1}{f(x)}}]:{\frac {1}{f(x)\phi (x)}}}
Le forme indeterminate del tipo
1
∞
,
∞
0
,
0
0
{\displaystyle \ 1^{\infty },\infty ^{0},0^{0}}
lim
x
→
c
log
f
(
x
)
ϕ
(
x
)
=
lim
x
→
c
ϕ
(
x
)
log
f
(
x
)
=
l
a
l
l
o
r
a
lim
x
→
c
f
(
x
)
ϕ
(
x
)
=
e
l
{\displaystyle \lim _{x\to \ c}\log f(x)^{\phi (x)}=\lim _{x\to c}\phi (x)\log f(x)=l\qquad allora\quad \lim _{x\to \ c}f(x)^{\phi (x)}=e^{l}}
formule di eulero sulle funzioni omogenee:
Se una funzione
z
=
f
(
x
,
y
)
{\displaystyle z=f(x,y)}
f
(
k
x
,
k
y
)
=
k
α
f
(
x
,
y
)
,
{\displaystyle \ f(kx,ky)=k^{\alpha }f(x,y),}
essendo
α
{\displaystyle \ \alpha }
a)
x
∂
f
∂
x
+
y
∂
f
∂
y
=
α
f
;
{\displaystyle x{\partial f \over \partial x}+y{\partial f \over \partial y}=\alpha f;}
b)
(
x
∂
f
∂
x
+
y
∂
f
∂
y
)
(
2
)
=
α
(
α
−
1
)
f
{\displaystyle (x{\partial f \over \partial x}+y{\partial f \over \partial y})^{(2)}=\alpha (\alpha -1)f}
....................
c)
(
x
∂
f
∂
x
+
y
∂
f
∂
y
)
(
r
)
=
α
(
α
−
1
)
.
.
.
(
α
−
r
+
1
)
f
.
{\displaystyle (x{\partial f \over \partial x}+y{\partial f \over \partial y})^{(r)}=\alpha (\alpha -1)...(\alpha -r+1)f.}
sulla ricerca delle radici di un'equazione: f(x)=0
a) Fra due radici di
f
(
x
)
=
0
{\displaystyle \ f(x)=0}
f
′
(
x
)
=
0
;
{\displaystyle \ f'(x)=0;}
b) condizione necessaria e sufficiente perché
α
{\displaystyle \ \alpha }
r
{\displaystyle \ r}
f
(
x
)
=
0
{\displaystyle \ f(x)=0}
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{\displaystyle \ f(\alpha )=f'(\alpha )=...=f^{(r-1)}(\alpha )=0,}
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{\displaystyle f^{(r)}\neq 0.}
![{\displaystyle \ P=[c,f(c)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53396a36004455e2a179a4c744850b28d5c1b75b)
![{\displaystyle \lim _{h\to 0}{f(x+h)-f(x) \over h}=\lim _{h\to 0}[{\ sen{h \over 2} \over {h \over 2}}\ cos(x+{h \over 2})]=\ cos\ x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4cbd6ca18611306d2bdcb0b759cf32ff4385494)
![{\displaystyle {\frac {d[cf(x)]}{dx}}=c\ {\frac {df}{dx}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92731a858f4b3c8a3c2ebde9ddce69fb6876a90c)
![{\displaystyle {\frac {d[f(x)]^{m}}{dx}}=m[f(x)]^{m-1}{\frac {df}{dx}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/286c2651d782be589ac221fb9bde2f5b523d3155)
![{\displaystyle {\frac {d{\sqrt[{n}]{x}}^{m}}{dx}}={\frac {m}{n}}{\sqrt[{n}]{x}}^{(m-n)};\qquad {\frac {d}{}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f3d7e3188bab5e31c85f55ccf991b12b42eea27)
![{\displaystyle {\frac {d^{n}f(x)}{dx^{n}}}={\frac {d}{dx}}[{\frac {d^{n-1}f(x)}{dx^{n-1}}}];\qquad d^{n}f={d^{n}f \over dx^{n}}(dx)^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3456694f608839920f9f5173eebaa08b2d8794)
![{\displaystyle f(x)\cdot \phi (x)={\frac {f(x)}{\frac {1}{\phi (x)}}}\qquad f(x)-\phi (x)=[{\frac {1}{\phi (x)}}-{\frac {1}{f(x)}}]:{\frac {1}{f(x)\phi (x)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5922ffd0070891b7b6745e7600ad6b4db0a52e58)
