This quick reference lists formulas often used in solving calculus problems.
Differentiation
edit
Wikipedia includes a useful summary of differentiation rules . These include rules for computing the derivatives of the sum, product, and other combinations of functions.
Rules for basic functions
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Here are the rules for the derivatives of the most common basic functions, where a is a real number.
Derivatives of powers
d
d
x
x
a
=
a
x
a
−
1
.
{\displaystyle {\frac {d}{dx}}x^{a}=ax^{a-1}.}
Exponential and logarithmic functions
d
d
x
e
x
=
e
x
.
{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}.}
d
d
x
a
x
=
a
x
ln
(
a
)
,
a
>
0
{\displaystyle {\frac {d}{dx}}a^{x}=a^{x}\ln(a),\qquad a>0}
d
d
x
ln
(
x
)
=
1
x
,
x
>
0.
{\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}},\qquad x>0.}
d
d
x
log
a
(
x
)
=
1
x
ln
(
a
)
,
x
,
a
>
0
{\displaystyle {\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}},\qquad x,a>0}
Trigonometric functions
d
d
x
sin
(
x
)
=
cos
(
x
)
.
{\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x).}
d
d
x
cos
(
x
)
=
−
sin
(
x
)
.
{\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x).}
d
d
x
tan
(
x
)
=
sec
2
(
x
)
=
1
cos
2
(
x
)
=
1
+
tan
2
(
x
)
.
{\displaystyle {\frac {d}{dx}}\tan(x)=\sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}=1+\tan ^{2}(x).}
Inverse trigonometric functions
d
d
x
arcsin
(
x
)
=
1
1
−
x
2
,
−
1
<
x
<
1.
{\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}},\qquad -1<x<1.}
d
d
x
arccos
(
x
)
=
−
1
1
−
x
2
,
−
1
<
x
<
1.
{\displaystyle {\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}},\qquad -1<x<1.}
d
d
x
arctan
(
x
)
=
1
1
+
x
2
{\displaystyle {\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}}
