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User:R36
Differentiation formulas
(Assume
f
(
x
)
{\displaystyle f(x)}
and
g
(
x
)
{\displaystyle g(x)}
exist,
a
{\displaystyle a}
and
c
{\displaystyle c}
are constants)
Integration formulas
1.
y
=
c
⋅
x
n
→
d
y
d
x
=
c
⋅
n
x
n
−
1
{\displaystyle y=c\cdot x^{n}\rightarrow {\frac {dy}{dx}}=c\cdot nx^{n-1}}
2.
y
=
f
(
x
)
⋅
g
(
x
)
→
d
y
d
x
=
f
(
x
)
⋅
g
′
(
x
)
+
g
(
x
)
⋅
f
′
(
x
)
{\displaystyle y=f(x)\cdot g(x)\rightarrow {\frac {dy}{dx}}=f(x)\cdot g'(x)+g(x)\cdot f'(x)}
20.
∫
u
n
d
u
=
u
n
+
1
n
+
1
+
c
(
if
n
≠
1
)
{\displaystyle \int u^{n}\,du={\frac {u^{n+1}}{n+1}}+c~({\mbox{if}}~n\neq 1)}
21.
∫
d
u
u
=
ln
|
u
|
+
c
{\displaystyle \int {\frac {du}{u}}=\ln |u|+c}
3.
y
=
f
(
x
)
g
(
x
)
→
d
y
d
x
=
g
(
x
)
⋅
f
′
(
x
)
−
f
(
x
)
⋅
g
′
(
x
)
[
g
(
x
)
]
2
{\displaystyle y={\frac {f(x)}{g(x)}}\rightarrow {\frac {dy}{dx}}={\frac {g(x)\cdot f'(x)-f(x)\cdot g'(x)}{[g(x)]^{2}}}}
4.
y
=
[
f
(
x
)
]
n
→
d
y
d
x
=
n
⋅
[
f
(
x
)
]
n
−
1
⋅
f
′
(
x
)
{\displaystyle y=[f(x)]^{n}\rightarrow {\frac {dy}{dx}}=n\cdot [f(x)]^{n-1}\cdot f'(x)}
22.
∫
sin
u
d
u
=
−
cos
u
+
c
{\displaystyle \int \sin u\,du=-\cos u+c}
23.
∫
cos
u
d
u
=
sin
u
+
c
{\displaystyle \int \cos u\,du=\sin u+c}
5.
y
=
sin
f
(
x
)
→
d
y
d
x
=
[
cos
f
(
x
)
]
⋅
f
′
(
x
)
{\displaystyle y=\sin f(x)\rightarrow {\frac {dy}{dx}}=[\cos f(x)]\cdot f'(x)}
6.
y
=
cos
f
(
x
)
→
d
y
d
x
=
[
−
sin
f
(
x
)
]
⋅
f
′
(
x
)
{\displaystyle y=\cos f(x)\rightarrow {\frac {dy}{dx}}=[-\sin f(x)]\cdot f'(x)}
24.
∫
tan
u
d
u
=
−
ln
|
cos
u
|
+
c
{\displaystyle \int \tan u\,du=-\ln |\cos u|+c}
25.
∫
cot
u
d
u
=
ln
|
sin
u
|
+
c
{\displaystyle \int \cot u\,du=\ln |\sin u|+c}
7.
y
=
tan
f
(
x
)
→
d
y
d
x
=
[
sec
2
f
(
x
)
]
⋅
f
′
(
x
)
{\displaystyle y=\tan f(x)\rightarrow {\frac {dy}{dx}}=[\sec ^{2}f(x)]\cdot f'(x)}
8.
y
=
cot
f
(
x
)
→
d
y
d
x
=
[
−
csc
2
f
(
x
)
]
⋅
f
′
(
x
)
{\displaystyle y=\cot f(x)\rightarrow {\frac {dy}{dx}}=[-\csc ^{2}f(x)]\cdot f'(x)}
26.
∫
sec
2
u
d
u
=
tan
u
+
c
{\displaystyle \int \sec ^{2}u\,du=\tan u+c}
27.
∫
csc
2
u
d
u
=
−
cot
u
+
c
{\displaystyle \int \csc ^{2}u\,du=-\cot u+c}
9.
y
=
sec
f
(
x
)
→
d
y
d
x
=
[
sec
f
(
x
)
⋅
tan
f
(
x
)
]
⋅
f
′
(
x
)
{\displaystyle y=\sec f(x)\rightarrow {\frac {dy}{dx}}=[\sec f(x)\cdot \tan f(x)]\cdot f'(x)}
10.
y
=
csc
f
(
x
)
→
d
y
d
x
=
[
−
csc
f
(
x
)
⋅
cot
f
(
x
)
]
⋅
f
′
(
x
)
{\displaystyle y=\csc f(x)\rightarrow {\frac {dy}{dx}}=[-\csc f(x)\cdot \cot f(x)]\cdot f'(x)}
28.
∫
sec
u
⋅
tan
u
d
u
=
sec
u
+
c
{\displaystyle \int \sec u\cdot \tan u\,du=\sec u+c}
29.
∫
csc
u
⋅
cot
u
d
u
=
−
csc
u
+
c
{\displaystyle \int \csc u\cdot \cot u\,du=-\csc u+c}
11.
y
=
ln
f
(
x
)
→
d
y
d
x
=
f
′
(
x
)
f
(
x
)
{\displaystyle y=\ln f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{f(x)}}}
12.
y
=
log
a
f
(
x
)
→
d
y
d
x
=
f
′
(
x
)
f
(
x
)
⋅
ln
a
{\displaystyle y=\log _{a}f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{f(x)\cdot \ln a}}}
30.
∫
sec
u
d
u
=
ln
|
sec
u
+
tan
u
|
+
c
{\displaystyle \int \sec u\,du=\ln |\sec u+\tan u|+c}
31.
∫
csc
u
d
u
=
−
ln
|
csc
u
+
cot
u
|
+
c
{\displaystyle \int \csc u\,du=-\ln |\csc u+\cot u|+c}
13.
y
=
e
f
(
x
)
→
d
y
d
x
=
e
f
(
x
)
⋅
f
′
(
x
)
{\displaystyle y=e^{f(x)}\rightarrow {\frac {dy}{dx}}=e^{f(x)}\cdot f'(x)}
14.
y
=
sin
−
1
f
(
x
)
→
d
y
d
x
=
f
′
(
x
)
1
−
[
f
(
x
)
]
2
{\displaystyle y=\sin ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{\sqrt {1-[f(x)]^{2}}}}}
32.
∫
e
u
d
u
=
e
u
+
c
{\displaystyle \int e^{u}\,du=e^{u}+c}
33.
∫
a
u
d
u
=
a
u
ln
a
+
c
{\displaystyle \int a^{u}\,du={\frac {a^{u}}{\ln a}}+c}
15.
y
=
cos
−
1
f
(
x
)
→
d
y
d
x
=
−
f
′
(
x
)
1
−
[
f
(
x
)
]
2
{\displaystyle y=\cos ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {-f'(x)}{\sqrt {1-[f(x)]^{2}}}}}
16.
y
=
tan
−
1
f
(
x
)
→
d
y
d
x
=
f
′
(
x
)
1
+
[
f
(
x
)
]
2
{\displaystyle y=\tan ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{1+[f(x)]^{2}}}}
34.
∫
d
u
a
2
−
u
2
=
sin
−
1
u
a
+
c
(
if
a
>
0
)
{\displaystyle \int {\frac {du}{\sqrt {a^{2}-u^{2}}}}=\sin ^{-1}{\frac {u}{a}}+c~({\mbox{if}}~a>0)}
35.
∫
d
u
a
2
−
u
2
=
1
a
tan
−
1
u
a
+
c
(
if
a
≠
0
)
{\displaystyle \int {\frac {du}{a^{2}-u^{2}}}={\frac {1}{a}}\tan ^{-1}{\frac {u}{a}}+c~({\mbox{if}}~a\neq 0)}
17.
y
=
cot
−
1
f
(
x
)
→
d
y
d
x
=
−
f
′
(
x
)
1
+
[
f
(
x
)
]
2
{\displaystyle y=\cot ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {-f'(x)}{1+[f(x)]^{2}}}}
18.
y
=
sec
−
1
f
(
x
)
→
d
y
d
x
=
f
′
(
x
)
f
(
x
)
⋅
[
f
(
x
)
]
2
−
1
{\displaystyle y=\sec ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{f(x)\cdot {\sqrt {[f(x)]^{2}-1}}}}}
36.
∫
d
u
u
u
2
−
a
2
=
1
a
sec
−
1
u
a
+
c
(
if
a
>
0
)
{\displaystyle \int {\frac {du}{u{\sqrt {u^{2}-a^{2}}}}}={\frac {1}{a}}\sec ^{-1}{\frac {u}{a}}+c~({\mbox{if}}~a>0)}
19.
y
=
csc
−
1
f
(
x
)
→
d
y
d
x
=
−
f
′
(
x
)
f
(
x
)
⋅
[
f
(
x
)
]
2
−
1
{\displaystyle y=\csc ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {-f'(x)}{f(x)\cdot {\sqrt {[f(x)]^{2}-1}}}}}
37. Integration by Parts
∫
u
d
v
=
u
v
−
∫
v
d
u
{\displaystyle \int u\,dv=uv-\int v\,du}
and 
exist, 
and 
are constants)
![{\displaystyle y={\frac {f(x)}{g(x)}}\rightarrow {\frac {dy}{dx}}={\frac {g(x)\cdot f'(x)-f(x)\cdot g'(x)}{[g(x)]^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/667fde943a0442dfaccd840884a093d74c00ea4e)
![{\displaystyle y=[f(x)]^{n}\rightarrow {\frac {dy}{dx}}=n\cdot [f(x)]^{n-1}\cdot f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8a30be1ed55e141626004d939c1b4464f5ff7e)
![{\displaystyle y=\sin f(x)\rightarrow {\frac {dy}{dx}}=[\cos f(x)]\cdot f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e931e5431a9413df2f8f29d820dfe6c3cb8312c0)
![{\displaystyle y=\cos f(x)\rightarrow {\frac {dy}{dx}}=[-\sin f(x)]\cdot f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/255ad121483f53ccc0a825e9d5fff3ab643a27da)
![{\displaystyle y=\tan f(x)\rightarrow {\frac {dy}{dx}}=[\sec ^{2}f(x)]\cdot f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ccaf879baf74216a7bf276db11fd4ee8bafc91)
![{\displaystyle y=\cot f(x)\rightarrow {\frac {dy}{dx}}=[-\csc ^{2}f(x)]\cdot f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d43beaca37e40d327f87363751d024b37b4c3aa2)
![{\displaystyle y=\sec f(x)\rightarrow {\frac {dy}{dx}}=[\sec f(x)\cdot \tan f(x)]\cdot f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/209811f64a585ea828e2a8717ab12d9d635a5725)
![{\displaystyle y=\csc f(x)\rightarrow {\frac {dy}{dx}}=[-\csc f(x)\cdot \cot f(x)]\cdot f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c26dc33c276946e14a7edd4bac5f75cbc9515d0)
![{\displaystyle y=\sin ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{\sqrt {1-[f(x)]^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0f2464ea8c43264b6a82f29dfa3fce65c77be4)
![{\displaystyle y=\cos ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {-f'(x)}{\sqrt {1-[f(x)]^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa71da694a9f5408120cab395fe472d2479a7d0)
![{\displaystyle y=\tan ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{1+[f(x)]^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08d43445eccedfa76c6c5361d5208613ed57036b)
![{\displaystyle y=\cot ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {-f'(x)}{1+[f(x)]^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1a2fc007e3bb3fb0d8d47068ccd64313eb5aa5)
![{\displaystyle y=\sec ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {f'(x)}{f(x)\cdot {\sqrt {[f(x)]^{2}-1}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc37be9c9cebf982ef5608ae25e2551c15b01c62)
![{\displaystyle y=\csc ^{-1}f(x)\rightarrow {\frac {dy}{dx}}={\frac {-f'(x)}{f(x)\cdot {\sqrt {[f(x)]^{2}-1}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bac3bd92e032a75db28c080a77ef1c8687f48fcb)
