यह कुछ मानक अवकल है:
अवकल की तालिका
d
d
x
c
=
0
{\displaystyle {d \over dx}c=0}
d
d
x
x
=
1
{\displaystyle {d \over dx}x=1}
d
d
x
c
x
=
c
{\displaystyle {d \over dx}cx=c}
d
d
x
|
x
|
=
x
|
x
|
=
sgn
x
,
x
≠
0
{\displaystyle {d \over dx}|x|={x \over |x|}=\operatorname {sgn} x,\qquad x\neq 0}
d
d
x
x
c
=
c
x
c
−
1
{\displaystyle {d \over dx}x^{c}=cx^{c-1}}
जहां दोनों xc and cxc-1 परिभाषित किया गया हैं।
d
d
x
(
1
x
)
=
d
d
x
(
x
−
1
)
=
−
x
−
2
=
−
1
x
2
{\displaystyle {d \over dx}\left({1 \over x}\right)={d \over dx}\left(x^{-1}\right)=-x^{-2}=-{1 \over x^{2}}}
d
d
x
(
1
x
c
)
=
d
d
x
(
x
−
c
)
=
−
c
x
c
+
1
{\displaystyle {d \over dx}\left({1 \over x^{c}}\right)={d \over dx}\left(x^{-c}\right)=-{c \over x^{c+1}}}
d
d
x
x
=
d
d
x
x
1
2
=
1
2
x
−
1
2
=
1
2
x
{\displaystyle {d \over dx}{\sqrt {x}}={d \over dx}x^{1 \over 2}={1 \over 2}x^{-{1 \over 2}}={1 \over 2{\sqrt {x}}}}
x > 0
d
d
x
c
x
=
c
x
ln
c
{\displaystyle {d \over dx}c^{x}={c^{x}\ln c}}
c > 0</math>
d
d
x
e
x
=
e
x
{\displaystyle {d \over dx}e^{x}=e^{x}}
d
d
x
log
c
x
=
1
x
ln
c
{\displaystyle {d \over dx}\log _{c}x={1 \over x\ln c}}
c > 0,
c
≠
1
{\displaystyle c\neq 1}
d
d
x
ln
x
=
1
x
{\displaystyle {d \over dx}\ln x={1 \over x}}
d
d
x
sin
x
=
cos
x
{\displaystyle {d \over dx}\sin x=\cos x}
d
d
x
cos
x
=
−
sin
x
{\displaystyle {d \over dx}\cos x=-\sin x}
d
d
x
tan
x
=
sec
2
x
{\displaystyle {d \over dx}\tan x=\sec ^{2}x}
d
d
x
sec
x
=
tan
x
sec
x
{\displaystyle {d \over dx}\sec x=\tan x\sec x}
d
d
x
cot
x
=
−
csc
2
x
{\displaystyle {d \over dx}\cot x=-\csc ^{2}x}
d
d
x
csc
x
=
−
csc
x
cot
x
{\displaystyle {d \over dx}\csc x=-\csc x\cot x}
d
d
x
arcsin
x
=
1
1
−
x
2
{\displaystyle {d \over dx}\arcsin x={1 \over {\sqrt {1-x^{2}}}}}
d
d
x
arccos
x
=
−
1
1
−
x
2
{\displaystyle {d \over dx}\arccos x={-1 \over {\sqrt {1-x^{2}}}}}
d
d
x
arctan
x
=
1
1
+
x
2
{\displaystyle {d \over dx}\arctan x={1 \over 1+x^{2}}}
d
d
x
arcsec
x
=
1
|
x
|
x
2
−
1
{\displaystyle {d \over dx}\operatorname {arcsec} x={1 \over |x|{\sqrt {x^{2}-1}}}}
d
d
x
arccot
x
=
−
1
1
+
x
2
{\displaystyle {d \over dx}\operatorname {arccot} x={-1 \over 1+x^{2}}}
d
d
x
arccsc
x
=
−
1
|
x
|
x
2
−
1
{\displaystyle {d \over dx}\operatorname {arccsc} x={-1 \over |x|{\sqrt {x^{2}-1}}}}
d
d
x
sinh
x
=
cosh
x
{\displaystyle {d \over dx}\sinh x=\cosh x}
d
d
x
cosh
x
=
sinh
x
{\displaystyle {d \over dx}\cosh x=\sinh x}
d
d
x
tanh
x
=
sech
2
x
{\displaystyle {d \over dx}\tanh x={\mbox{sech}}^{2}x}
d
d
x
sech
x
=
−
tanh
x
sech
x
{\displaystyle {d \over dx}{\mbox{sech}}x=-\tanh x{\mbox{sech}}x}
d
d
x
coth
x
=
−
csch
2
x
{\displaystyle {d \over dx}{\mbox{coth}}x=-{\mbox{csch}}^{2}x}
d
d
x
csch
x
=
−
coth
x
csch
x
{\displaystyle {d \over dx}{\mbox{csch}}x=-{\mbox{coth}}x{\mbox{csch}}x}
d
d
x
arcsinh
x
=
1
x
2
+
1
{\displaystyle {d \over dx}{\mbox{arcsinh}}x={1 \over {\sqrt {x^{2}+1}}}}
d
d
x
arccosh
x
=
1
x
2
−
1
{\displaystyle {d \over dx}{\mbox{arccosh}}x={1 \over {\sqrt {x^{2}-1}}}}
d
d
x
arctanh
x
=
1
1
−
x
2
{\displaystyle {d \over dx}{\mbox{arctanh}}x={1 \over 1-x^{2}}}
d
d
x
arcsech
x
=
1
x
1
−
x
2
{\displaystyle {d \over dx}{\mbox{arcsech}}x={1 \over x{\sqrt {1-x^{2}}}}}
d
d
x
arccoth
x
=
1
1
−
x
2
{\displaystyle {d \over dx}{\mbox{arccoth}}x={1 \over 1-x^{2}}}
d
d
x
arccsch
x
=
−
1
|
x
|
1
+
x
2
{\displaystyle {d \over dx}{\mbox{arccsch}}x={-1 \over |x|{\sqrt {1+x^{2}}}}}
[ सम्पादन ] किसी दिए गए फ़ंक्शन को बार-बार अवकल करने की प्रक्रिया को क्रमिक अवकल कहा जाता है। वैज्ञानिक और इंजीनियरिंग अनुप्रयोगों के सभी क्षेत्रों में फ़ंक्शन के प्रसार के लिये उच्च क्रम (higher order) अवकल गुणांक आते रहते हैं।
y
=
f
(
x
)
{\displaystyle y=f(x)}
1.
y
1
=
d
y
d
x
{\displaystyle y_{1}={\operatorname {d} \!y \over \operatorname {d} \!x}}
y
2
=
d
y
2
d
x
2
{\displaystyle y_{2}={\operatorname {d} \!y^{2} \over \operatorname {d} \!x^{2}}}
y
3
=
d
y
3
d
x
3
{\displaystyle y_{3}={\operatorname {d} \!y^{3} \over \operatorname {d} \!x^{3}}}
y
n
=
d
y
n
d
x
n
{\displaystyle y_{n}={\operatorname {d} \!y^{n} \over \operatorname {d} \!x^{n}}}
2.
f
(
x
)
′
,
f
′
(
x
)
,
f
″
(
x
)
,
f
(
3
)
(
x
)
{\displaystyle f(x)^{\prime },f'(x),f''(x),f^{(3)}(x)}
f
(
n
)
(
x
)
{\displaystyle f^{(n)}(x)}
y
=
s
i
n
−
1
(
x
)
{\displaystyle y=sin^{-1}(x)}
(
1
−
x
2
)
d
y
2
d
x
2
−
x
.
d
y
d
x
=
0
{\displaystyle (1-x^{2}){\operatorname {d} \!y^{2} \over \operatorname {d} \!x^{2}}-x.{\operatorname {d} \!y \over \operatorname {d} \!x}=0}
