Chapter 3
Derivatives of Elementary
Functions
3.1 DERIVATIVES OF ALGEBRAIC, LOGARITHMIC,
AND EXPONENTIAL FUNCTIONS
3.1.1
Let u(x) be a differentiable function with respect to x, and α, a and k be constants.
1.
d α
[x ] = αxα−1
dx
2.
du
d α
[u ] = αuα−1
dx
dx
3.
1
d 1/2
[x ] = 1/2
dx
2x
4.
d 1/2
1 du
[u ] = 1/2
dx
2u dx
5.
d −α
α
[x ] = − α+1
dx
x
6.
d −α
α du
[u ] = − α+1
dx
u
dx
7.
d
1
[ln x] =
dx
x
8.
d
1 du
[ln u] =
dx
u dx
9.
(n − 1)!
(−1)
dn
[ln x] =
dxn
xn
10.
d2
1
[ln u] = − 2
dx2
u
11.
d
[x ln x] = ln x + 1
dx
12.
d
x du
[x ln u] = ln u +
dx
u dx
13.
d n
[x ln x] = (n ln x + 1)xn−1
dx
14.
d
du
[u ln u] = (ln u + 1)
dx
dx
n−1
149
du
dx
2
+
1 d2 u
u dx2
150
Chapter 3
15.
d x
[e ] = ex
dx
16.
d kx
[e ] = kekx
dx
17.
du
d u
[e ] = eu
dx
dx
18.
d x
[a ] = ax ln a
dx
19.
d u
du
[a ] = au (ln a)
dx
dx
20.
d x
[x ] = (1 + ln x)xx
dx
Derivatives of Elementary Functions
3.2 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
3.2.1
Let u(x) be a differentiable function with respect to x.
1.
d
[sin x] = cos x
dx
2.
d
du
[sin u] = cos u
dx
dx
3.
d
[cos x] = −sin x
dx
4.
d
du
[cos u] = −sin u
dx
dx
5.
d
[tan x] = sec2 x
dx
6.
d
du
[tan u] = sec2 u
dx
dx
7.
d
[csc x] = −csc x cot x
dx
8.
d
du
[csc u] = −csc u cot u
dx
dx
9.
d
[sec x] = sec x tan x
dx
11.
13.
d
[cot x] = −csc2 x
dx
1
dn
[sin
x]
=
sin
x
+
nπ
dxn
2
10.
12.
14.
d
du
[sec u] = sec u tan u
dx
dx
d
du
[cot u] = −csc2 u
dx
dx
1
dn
[cos
x]
=
cos
x
+
nπ
dxn
2
3.3 DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS
3.3.1
Let u(x) be a differentiable function with respect to x.
1.
2.
3.
4.
π
d x
1
x
π
arcsin
=
−
<
arcsin
<
1/2
dx
a
2
a
2
(a2 − x2 )
d
u
1
π
du
u
π
arcsin
=
−
<
arcsin
<
1/2
dx
a
2
a
2
(a2 − u2 ) dx
x
−1
x
d
arccos
=
0 < arccos < π
1/2
2
2
dx
a
a
(a − x )
u
−1
u
du
d
arccos
=
0
<
arccos
<
π
1/2
dx
a
a
(a2 − u2 ) dx
3.4 Derivatives of Hyperbolic Functions
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
151
x
a
d arctan
= 2
dx
a
a + x2
du
u
a
d arctan
= 2
2
dx
a
a + u dx
x
d x
−a
π
0
<
arccsc
arccsc
=
<
1/2
dx
a
a
2
x (x2 − a2 )
d
x
a
π
x
arccsc
=
− < arccsc < 0
1/2
dx
a
2
a
x (x2 − a2 )
d
u
−a
u
du
π
arccsc
=
0 < arccsc <
1/2
dx
a
a
2
u (u2 − a2 ) dx
u
a
π
d
du
u
arccsc
=
− < arccsc < 0
1/2
dx
a
2
a
u (u2 − a2 ) dx
x
a
x
π
d
arcsec
=
0
<
arcsec
<
1/2
dx
a
a
2
x (x2 − a2 )
x
−a
π
d
x
arcsec
=
<
arcsec
<
π
1/2
dx
a
2
a
x (x2 − a2 )
u
a
u
du
π
d
arcsec
=
0
<
arcsec
<
1/2
dx
a
a
2
u (u2 − a2 ) dx
u
−a
π
d
du
u
arcsec
=
<
arcsec
<
π
1/2
dx
a
2
a
u (u2 − a2 ) dx
3.4 DERIVATIVES OF HYPERBOLIC FUNCTIONS
3.4.1
Let u(x) be a differentiable function with respect to x.
1.
d
[sinh x] = cosh x
dx
2.
d
du
[sinh u] = cosh u
dx
dx
3.
d
[cosh x] = sinh x
dx
4.
d
du
[cosh u] = sinh u
dx
dx
5.
d
[tanh x] = sech2 x
dx
6.
d
du
[tanh u] = sech2 u
dx
dx
7.
d
[csch x] = − csch x coth x
dx
8.
d
du
[csch u] = − csch u coth u
dx
dx
9.
d
[sech x] = − sech x tanh x
dx
10.
d
du
[sech u] = − sech u tanh u
dx
dx
d
[coth x] = − csch2 x + 2δ(x)
dx
12.
d
du
[coth u] = −[csch2 u + 2δ(u)]
dx
dx
11.
The delta function occurs because of the discontinuity in the coth function at the origin.
152
Chapter 3
Derivatives of Elementary Functions
3.5 DERIVATIVES OF INVERSE HYPERBOLIC FUNCTIONS
3.5.1
Let u(x) be a differentiable function with respect to x
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
x
1
d arcsinh
=
1/2
2
dx
a
(x + a2 )
d
u
1
du
arcsinh
=
1/2 dx
2
2
dx
a
(u + a )
x
d x
1
x
arccosh
=
>
1,
arccosh
>
0
1/2
dx
a
a
a
(x2 − a2 )
d
x
−1
x
x
arccosh
=
>
1,
arccosh
<
0
1/2
dx
a
a
a
(x2 − a2 )
u
1
u
du
u
d
arccosh
=
>
1,
arccosh
>
0
1/2
dx
a
a
a
(u2 − a2 ) dx
u
−1
du u
u
d
arccosh
=
>
1,
arccosh
<
0
1/2
dx
a
a
a
(u2 − a2 ) dx
2
x
a
d
x < a2
arctanh
= 2
dx
a
a − x2
du 2
u
a
d arctanh
= 2
u < a2
2
dx
a
a − u dx
x
−a
d arccsch
=
[x = 0]
1/2
dx
a
|x| (x2 + a2 )
du
d u
−a
arccsch
=
[u = 0]
1/2
dx
a
|u| (u2 + a2 ) dx
d x
−a
x
x
arcsech
=
0
<
<
1,
arcsech
>
0
1/2
dx
a
a
a
x (a2 − x2 )
d
x
a
x
x
arcsech
=
0 < < 1, arcsech < 0
1/2
dx
a
a
a
x (a2 − x2 )
u
−a
u
du
u
d
arcsech
=
0
<
<
1,
arcsech
>
0
1/2
dx
a
a
a
u (a2 − u2 ) dx
u
a
u
du
u
d
arcsech
=
0
<
<
1,
arcsech
<
0
1/2
dx
a
a
a
u (a2 − u2 ) dx
2
d
x
a
x > a2
arccoth
= 2
2
dx
a
a −x
d du 2
u
a
arccoth
= 2
u > a2
2
dx
a
a − u dx