The derivative of any polynomial function of one variable is easily obtained. If
c
∈
R
{\displaystyle c\in\R}
f
,
g
:
D
→
R
{\displaystyle f,g:D\to\R}
D
′
{\displaystyle D'}
c
⋅
f
{\displaystyle c\cdot f}
f
+
g
{\displaystyle f+g}
f
⋅
g
{\displaystyle f \cdot g}
g
{\displaystyle g}
D
′
{\displaystyle D'}
1
g
{\displaystyle \frac{1}{g}}
f
g
{\displaystyle \frac{f}{g}}
D
′
{\displaystyle D'}
f
{\displaystyle f}
g
(
D
′
)
{\displaystyle g\left(D'\right)}
f
∘
g
{\displaystyle f \circ g}
D
′
{\displaystyle D'}
f
(
x
)
=
a
{\displaystyle f(x)=a}
a
{\displaystyle a}
a
′
=
0
{\displaystyle a'=0}
[Proof] For any
r
∈
R
{\displaystyle r \in \R}
(
x
r
)
′
=
r
x
r
−
1
{\displaystyle (x^r)'=rx^{r-1}}
[Proof] Which covers any single variable polynomial function. Derivatives of non-polynomial functions require additional rules.
For any real-valued differentiable functions
f
(
x
)
,
g
(
x
)
{\displaystyle f(x), g(x)}
The definition of the derivative:
f
′
(
x
)
=
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
{\displaystyle f'(x) = \lim_{h \rightarrow 0} \dfrac{f(x+h) -f(x)}{h}}
Basic derivative rules:
(
a
⋅
f
(
x
)
)
′
=
a
⋅
f
′
(
x
)
{\displaystyle (a\cdot f(x))'=a\cdot f'(x)}
[Proof]
(
f
(
x
)
±
g
(
x
)
)
′
=
f
′
(
x
)
±
g
′
(
x
)
{\displaystyle (f(x)\pm g(x))'=f'(x)\pm g'(x)}
[Proof]
(
−
f
(
x
)
)
′
=
−
f
′
(
x
)
{\displaystyle (-f(x))'=-f'(x)}
(
f
(
x
)
⋅
g
(
x
)
)
′
=
f
′
(
x
)
g
(
x
)
+
f
(
x
)
g
′
(
x
)
{\displaystyle (f(x)\cdot g(x))' =f'(x)g(x)+f(x)g'(x)}
Product rule )
(
f
(
x
)
g
(
x
)
)
′
=
f
′
(
x
)
g
(
x
)
−
f
(
x
)
g
′
(
x
)
g
(
x
)
2
{\displaystyle \left(\frac{f(x)}{g(x)}\right)'=\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^2}}
Quotient rule )
(
1
g
(
x
)
)
′
=
−
g
′
(
x
)
g
(
x
)
2
{\displaystyle \left(\frac{1}{g(x)}\right)'=-\frac{g'(x)}{g(x)^2}}
(
f
∘
g
)
′
=
(
f
(
g
(
x
)
)
)
′
=
(
f
′
(
g
)
)
⋅
g
′
(
x
)
{\displaystyle (f\circ g)'=\bigl(f(g(x))\bigr)'=(f'(g))\cdot g'(x)}
Chain rule )Trigonometric functions :
d
d
x
(
sin
(
x
)
)
=
cos
(
x
)
{\displaystyle \frac{d}{dx}(\sin(x))=\cos(x)}
[Proof]
d
d
x
(
cos
(
x
)
)
=
−
sin
(
x
)
{\displaystyle \frac{d}{dx}(\cos(x))=-\sin(x)}
d
d
x
(
tan
(
x
)
)
=
sec
2
(
x
)
{\displaystyle \frac{d}{dx}(\tan(x))=\sec^2(x)}
d
d
x
(
csc
(
x
)
)
=
−
csc
(
x
)
cot
(
x
)
{\displaystyle \frac{d}{dx}(\csc(x))=-\csc(x)\cot(x)}
d
d
x
(
sec
(
x
)
)
=
sec
(
x
)
tan
(
x
)
{\displaystyle \frac{d}{dx}(\sec(x))=\sec(x)\tan(x)}
d
d
x
(
cot
(
x
)
)
=
−
csc
2
(
x
)
{\displaystyle \frac{d}{dx}(\cot(x))=-\csc^2(x)}
d
d
x
(
arcsin
(
x
)
)
=
1
1
−
x
2
{\displaystyle \frac{d}{dx}(\arcsin(x))=\frac{1}{\sqrt{1-x^2}}}
Proof ]
d
d
x
(
arccos
(
x
)
)
=
−
1
1
−
x
2
{\displaystyle \frac{d}{dx}(\arccos(x))=-\frac{1}{\sqrt{1-x^2}}}
d
d
x
(
arctan
(
x
)
)
=
1
1
+
x
2
{\displaystyle \frac{d}{dx}(\arctan(x))=\frac{1}{1+x^2}}
d
d
x
(
arcsec
(
x
)
)
=
1
|
x
|
x
2
−
1
{\displaystyle {\frac {d}{dx}}(\operatorname {arcsec}(x))={\frac {1}{|x|{\sqrt {x^{2}-1}}}}}
d
d
x
(
arccsc
(
x
)
)
=
−
1
|
x
|
x
2
−
1
{\displaystyle {\frac {d}{dx}}(\operatorname {arccsc}(x))=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}}
d
d
x
(
arccot
(
x
)
)
=
−
1
1
+
x
2
{\displaystyle {\frac {d}{dx}}(\operatorname {arccot}(x))=-{\frac {1}{1+x^{2}}}}
Logarithmic and exponential functions :
d
d
x
(
e
x
)
=
e
x
{\displaystyle \frac{d}{dx}(e^x)=e^x}
Proof ]
d
d
x
(
a
x
)
=
a
x
ln
(
a
)
{\displaystyle \frac{d}{dx}(a^x)=a^x\ln(a)}
[Proof]
d
d
x
(
ln
(
|
x
|
)
)
=
1
x
{\displaystyle \frac{d}{dx}(\ln(|x|))=\frac{1}{x}}
[Proof]
d
d
x
(
log
a
(
x
)
)
=
1
ln
(
a
)
x
{\displaystyle \frac{d}{dx}(\log_a(x))=\frac{1}{\ln(a)x}}
Proof ]Other:
(
f
−
1
(
x
)
)
′
=
1
f
′
(
f
−
1
(
x
)
)
{\displaystyle (f^{-1}(x))'={\frac {1}{f'(f^{-1}(x))}}}
Proof ]
d
d
x
(
f
(
x
)
g
(
x
)
)
=
(
g
′
(
x
)
ln
(
f
(
x
)
)
+
g
(
x
)
f
′
(
x
)
f
(
x
)
)
f
(
x
)
g
(
x
)
{\displaystyle \frac d{dx} (f(x)^{g(x)}) = \Big(g'(x)\ln(f(x)) + \frac {g(x)f'(x)}{f(x)} \Big)f(x)^{g(x)}}
Proof ]
