1.4.E8
f
(
2
)
(
x
)
=
d
2
f
d
x
2
superscript
𝑓
2
𝑥
derivative
𝑓
𝑥
2
{\displaystyle{\displaystyle f^{(2)}(x)=\frac{{\mathrm{d}}^{2}f}{{\mathrm{d}x}%
^{2}}}}
f^{ (2)} (x) = \deriv [2] { f}{ x} ( f ( x )) ^ ( 2 ) = diff ( f , [ x $ ( 2 )])
( f [ x ]) ^ ( 2 ) == D [ f , { x , 2 }]
Failure
Failure
Expand Failed [30 / 30]
Result : .7500000006 + 1.299038106 * I
Test Values : { f = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , x = 1.5 }
Result : .2500000002 + .4330127020 * I
Test Values : { f = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , x = .5 }
Result : 1.000000001 + 1.732050808 * I
Test Values : { f = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , x = 2 }
Result : -.7500000006-1.299038106 * I
Test Values : { f = -1 / 2 + 1 / 2 * I * 3 ^ ( 1 / 2 ), x = 1.5 }
... skip entries to safe data
Expand Failed [30 / 30]
Result : Complex [ 0.7500000000000002 , 1.299038105676658 ]
Test Values : { Rule [ f , Power [ E , Times [ Complex [ 0 , Rational [ 1 , 6 ]], Pi ]]], Rule [ x , 1.5 ]}
Result : Complex [ 0.25000000000000006 , 0.4330127018922193 ]
Test Values : { Rule [ f , Power [ E , Times [ Complex [ 0 , Rational [ 1 , 6 ]], Pi ]]], Rule [ x , 0.5 ]}
... skip entries to safe data
1.4.E8
d
2
f
d
x
2
=
d
d
x
(
d
f
d
x
)
derivative
𝑓
𝑥
2
derivative
𝑥
derivative
𝑓
𝑥
{\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}f}{{\mathrm{d}x}^{2}}=\frac{%
\mathrm{d}}{\mathrm{d}x}\left(\frac{\mathrm{d}f}{\mathrm{d}x}\right)}}
\deriv [2] { f}{ x} = \deriv {}{ x} \left (\deriv { f}{ x} \right )diff ( f , [ x $ ( 2 )]) = diff ( diff ( f , x ), x )
D [ f , { x , 2 }] == D [ D [ f , x ], x ]
Successful
Successful
Skip - symbolical successful subtest
Successful [Tested: 30]
1.4.E9
f
(
n
)
(
x
)
=
d
d
x
f
(
n
-
1
)
(
x
)
superscript
𝑓
𝑛
𝑥
derivative
𝑥
superscript
𝑓
𝑛
1
𝑥
{\displaystyle{\displaystyle f^{(n)}(x)=\frac{\mathrm{d}}{\mathrm{d}x}f^{(n-1)%
}(x)}}
f^{ (n)} (x) = \deriv {}{ x} f^{ (n-1)} (x)( f ( x )) ^ ( n ) = diff (( f ( x )) ^ ( n - 1 ), x )
( f [ x ]) ^ ( n ) == D [( f [ x ]) ^ ( n - 1 ), x ]
Failure
Failure
Expand Failed [84 / 90]
Result : .299038106 + .7500000000 * I
Test Values : { f = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , x = 1.5 , n = 1 }
Result : -.1160254034 + .7990381060 * I
Test Values : { f = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , x = 1.5 , n = 2 }
Result : -.4999999999 + .6339745980 * I
Test Values : { f = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , x = 1.5 , n = 3 }
Result : -.5669872980 + .2500000000 * I
Test Values : { f = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , x = .5 , n = 1 }
... skip entries to safe data
Expand Failed [84 / 90]
Result : Complex [ 0.299038105676658 , 0.7499999999999999 ]
Test Values : { Rule [ f , Power [ E , Times [ Complex [ 0 , Rational [ 1 , 6 ]], Pi ]]], Rule [ n , 1 ], Rule [ x , 1.5 ]}
Result : Complex [ -0.11602540378443849 , 0.799038105676658 ]
Test Values : { Rule [ f , Power [ E , Times [ Complex [ 0 , Rational [ 1 , 6 ]], Pi ]]], Rule [ n , 2 ], Rule [ x , 1.5 ]}
... skip entries to safe data
1.4.E16
∫
f
g
d
x
=
(
∫
f
d
x
)
g
-
∫
(
∫
f
d
x
)
d
g
d
x
d
x
𝑓
𝑔
𝑥
𝑓
𝑥
𝑔
𝑓
𝑥
derivative
𝑔
𝑥
𝑥
{\displaystyle{\displaystyle\int fg\mathrm{d}x=\left(\int f\mathrm{d}x\right)g%
-\int\left(\int f\mathrm{d}x\right)\frac{\mathrm{d}g}{\mathrm{d}x}\mathrm{d}x}}
\int fg\diff { x} = \left (\int f\diff { x} \right )g-\int\left (\int f\diff { x} \right )\deriv { g}{ x} \diff { x} int ( f * g , x ) = ( int ( f , x )) * g - int (( int ( f , x )) * diff ( g , x ), x )
Integrate [ f * g , x , GenerateConditions -> None ] == ( Integrate [ f , x , GenerateConditions -> None ]) * g - Integrate [( Integrate [ f , x , GenerateConditions -> None ]) * D [ g , x ], x , GenerateConditions -> None ]
Successful
Successful
-
Successful [Tested: 100]
1.4.E36
R
n
=
f
(
n
+
1
)
(
c
)
(
n
+
1
)
!
(
x
-
a
)
n
+
1
subscript
𝑅
𝑛
superscript
𝑓
𝑛
1
𝑐
𝑛
1
superscript
𝑥
𝑎
𝑛
1
{\displaystyle{\displaystyle R_{n}=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}}}
R_{ n} = \frac { f^{ (n+1)} (c)}{ (n+1)!} (x-a)^{ n+1}
a
<
c
,
c
<
x
formulae-sequence
𝑎
𝑐
𝑐
𝑥
{\displaystyle{\displaystyle a<c,c<x}}
R[n] = ((f(c))^(n + 1))/(factorial(n + 1))*(x - a)^(n + 1) Subscript[R, n] == Divide[(f[c])^(n + 1),(n + 1)!]*(x - a)^(n + 1) Skipped - no semantic math
Skipped - no semantic math
-
-
1.4.E37
R
n
=
1
n
!
∫
a
x
(
x
-
t
)
n
f
(
n
+
1
)
(
t
)
d
t
subscript
𝑅
𝑛
1
𝑛
subscript
superscript
𝑥
𝑎
superscript
𝑥
𝑡
𝑛
superscript
𝑓
𝑛
1
𝑡
𝑡
{\displaystyle{\displaystyle R_{n}=\frac{1}{n!}\int^{x}_{a}(x-t)^{n}f^{(n+1)}(%
t)\mathrm{d}t}}
R_{ n} = \frac { 1}{ n!} \int ^{ x}_{ a} (x-t)^{ n} f^{ (n+1)} (t)\diff { t} R [ n ] = ( 1 ) / ( factorial ( n )) * int (( x - t ) ^ ( n ) * ( f ( t )) ^ ( n + 1 ), t = a .. x )
Subscript [ R , n ] == Divide [ 1 ,( n ) ! ] * Integrate [( x - t ) ^ ( n ) * ( f [ t ]) ^ ( n + 1 ), { t , a , x }, GenerateConditions -> None ]
Failure
Failure
Expand Failed [300 / 300]
Result : 1.991025404 + 2.448557159 * I
Test Values : { a = -1.5 , f = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , x = 1.5 , R [ n ] = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , n = 1 }
Result : .8660254040 + 3.875000000 * I
Test Values : { a = -1.5 , f = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , x = 1.5 , R [ n ] = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , n = 2 }
Result : -.6527245960 + 3.130552164 * I
Test Values : { a = -1.5 , f = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , x = 1.5 , R [ n ] = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , n = 3 }
Result : .6250000000 + 2.814582563 * I
Test Values : { a = -1.5 , f = 1 / 2 * 3 ^ ( 1 / 2 ) + 1 / 2 * I , x = 1.5 , R [ n ] = -1 / 2 + 1 / 2 * I * 3 ^ ( 1 / 2 ), n = 1 }
... skip entries to safe data
Expand Failed [300 / 300]
Result : Complex [ 1.9910254037844388 , 2.4485571585149866 ]
Test Values : { Rule [ a , -1.5 ], Rule [ f , Power [ E , Times [ Complex [ 0 , Rational [ 1 , 6 ]], Pi ]]], Rule [ n , 1 ], Rule [ x , 1.5 ], Rule [ Subscript [ R , n ], Power [ E , Times [ Complex [ 0 , Rational [ 1 , 6 ]], Pi ]]]}
Result : Complex [ 0.8660254037844387 , 3.875 ]
Test Values : { Rule [ a , -1.5 ], Rule [ f , Power [ E , Times [ Complex [ 0 , Rational [ 1 , 6 ]], Pi ]]], Rule [ n , 2 ], Rule [ x , 1.5 ], Rule [ Subscript [ R , n ], Power [ E , Times [ Complex [ 0 , Rational [ 1 , 6 ]], Pi ]]]}
... skip entries to safe data