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Figure 1.15 : Estimation of the derivative, which is the slope of the tangent line. When point B approaches point A, the slope of the line AB approaches the slope of the tangent to the curve at point A.
This section provides a quick introduction to the idea of the derivative. For a more detailed discussion and exploration of the differentiation and of Calculus, see Calculus and Differentiation .
Often we are interested in the slope of a line tangent to a function
y
(
x
)
{\displaystyle y(x)}
x
{\displaystyle x}
d
y
/
d
x
{\displaystyle dy/dx}
x
{\displaystyle x}
x
{\displaystyle x}
g
(
x
)
=
d
y
(
x
)
d
x
.
{\displaystyle g(x)={\frac {dy(x)}{dx}}.}
As figure 1.15 illustrates, the slope of the tangent line at some point on the function may be approximated by the slope of a line connecting two points, A and B, set a finite distance apart on the curve:
d
y
d
x
≈
Δ
y
Δ
x
.
{\displaystyle {\frac {dy}{dx}}\approx {\frac {\Delta y}{\Delta x}}.}
As B is moved closer to A, the approximation becomes better. In the limit when B moves infinitely close to A, it is exact.
Derivatives of some common functions are now given. In each case
c
{\displaystyle c}
Table of Derivatives
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}}
where both x c cx c −1
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
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
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=\operatorname {sech} ^{2}x}
d
d
x
sech
x
=
−
tanh
x
sech
x
{\displaystyle {d \over dx}\operatorname {sech} x=-\tanh x\operatorname {sech} x}
d
d
x
coth
x
=
−
csch
2
x
{\displaystyle {d \over dx}\operatorname {coth} x=-\operatorname {csch} ^{2}x}
d
d
x
csch
x
=
−
coth
x
csch
x
{\displaystyle {d \over dx}\operatorname {csch} x=-\operatorname {coth} x\operatorname {csch} x}
d
d
x
arsinh
x
=
1
x
2
+
1
{\displaystyle {d \over dx}\operatorname {arsinh} x={1 \over {\sqrt {x^{2}+1}}}}
d
d
x
arcosh
x
=
1
x
2
−
1
{\displaystyle {d \over dx}\operatorname {arcosh} x={1 \over {\sqrt {x^{2}-1}}}}
d
d
x
artanh
x
=
1
1
−
x
2
{\displaystyle {d \over dx}\operatorname {artanh} x={1 \over 1-x^{2}}}
d
d
x
arsech
x
=
1
x
1
−
x
2
{\displaystyle {d \over dx}\operatorname {arsech} x={1 \over x{\sqrt {1-x^{2}}}}}
d
d
x
arcoth
x
=
1
1
−
x
2
{\displaystyle {d \over dx}\operatorname {arcoth} x={1 \over 1-x^{2}}}
d
d
x
arcsch
x
=
−
1
|
x
|
1
+
x
2
{\displaystyle {d \over dx}\operatorname {arcsch} x={-1 \over |x|{\sqrt {1+x^{2}}}}}
The product and chain rules are used to compute the derivatives of complex functions. For instance,
d
d
x
(
sin
(
x
)
cos
(
x
)
)
=
d
sin
(
x
)
d
x
cos
(
x
)
+
sin
(
x
)
d
cos
(
x
)
d
x
=
cos
2
(
x
)
−
sin
2
(
x
)
{\displaystyle {\frac {d}{dx}}(\sin(x)\cos(x))={\frac {d\sin(x)}{dx}}\cos(x)+\sin(x){\frac {d\cos(x)}{dx}}=\cos ^{2}(x)-\sin ^{2}(x)}
and
d
d
x
log
(
sin
(
x
)
)
=
1
sin
(
x
)
d
sin
(
x
)
d
x
=
cos
(
x
)
sin
(
x
)
.
{\displaystyle {\frac {d}{dx}}\log(\sin(x))={\frac {1}{\sin(x)}}{\frac {d\sin(x)}{dx}}={\frac {\cos(x)}{\sin(x)}}.}
