Elementary Functions - 4.34 Derivatives and Differential Equations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.34.E1 d d z ⁡ sinh ⁡ z = cosh ⁡ z derivative 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sinh z=\cosh z}}
\deriv{}{z}\sinh@@{z} = \cosh@@{z}		 

diff(sinh(z), z) = cosh(z)

D[Sinh[z], z] == Cosh[z]
Successful Successful - Successful [Tested: 7]
4.34.E2 d d z ⁡ cosh ⁡ z = sinh ⁡ z derivative 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cosh z=\sinh z}}
\deriv{}{z}\cosh@@{z} = \sinh@@{z}		 

diff(cosh(z), z) = sinh(z)

D[Cosh[z], z] == Sinh[z]
Successful Successful - Successful [Tested: 7]
4.34.E3 d d z ⁡ tanh ⁡ z = sech 2 ⁡ z derivative 𝑧 𝑧 2 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\tanh z={% \operatorname{sech}^{2}}z}}
\deriv{}{z}\tanh@@{z} = \sech^{2}@@{z}		 

diff(tanh(z), z) = (sech(z))^(2)

D[Tanh[z], z] == (Sech[z])^(2)
Successful Successful - Successful [Tested: 7]
4.34.E4 d d z ⁡ csch ⁡ z = - csch ⁡ z ⁢ coth ⁡ z derivative 𝑧 𝑧 𝑧 hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{csch}z% =-\operatorname{csch}z\coth z}}
\deriv{}{z}\csch@@{z} = -\csch@@{z}\coth@@{z}		 

diff(csch(z), z) = - csch(z)*coth(z)

D[Csch[z], z] == - Csch[z]*Coth[z]
Successful Successful - Successful [Tested: 7]
4.34.E5 d d z ⁡ sech ⁡ z = - sech ⁡ z ⁢ tanh ⁡ z derivative 𝑧 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sech}z% =-\operatorname{sech}z\tanh z}}
\deriv{}{z}\sech@@{z} = -\sech@@{z}\tanh@@{z}		 

diff(sech(z), z) = - sech(z)*tanh(z)

D[Sech[z], z] == - Sech[z]*Tanh[z]
Successful Successful - Successful [Tested: 7]
4.34.E6 d d z ⁡ coth ⁡ z = - csch 2 ⁡ z derivative 𝑧 hyperbolic-cotangent 𝑧 2 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\coth z=-{% \operatorname{csch}^{2}}z}}
\deriv{}{z}\coth@@{z} = -\csch^{2}@@{z}		 

diff(coth(z), z) = - (csch(z))^(2)

D[Coth[z], z] == - (Csch[z])^(2)
Successful Successful - Successful [Tested: 7]
4.34.E7 d 2 w d z 2 - a 2 ⁢ w = 0 derivative 𝑤 𝑧 2 superscript 𝑎 2 𝑤 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-a^{2}w% =0}}
\deriv[2]{w}{z}-a^{2}w = 0		 

diff(w, [z$(2)])- (a)^(2)* w = 0

D[w, {z, 2}]- (a)^(2)* w == 0
Failure Failure
Failed [300 / 300]
Result: -1.948557159-1.125000000*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -1.948557159-1.125000000*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: -1.948557159-1.125000000*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}


Result: -1.948557159-1.125000000*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.948557158514987, -1.1249999999999998]
Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.948557158514987, -1.1249999999999998]
Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.34.E8 ( d w d z ) 2 - a 2 ⁢ w 2 = 1 superscript derivative 𝑤 𝑧 2 superscript 𝑎 2 superscript 𝑤 2 1 {\displaystyle{\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a% ^{2}w^{2}=1}}
\left(\deriv{w}{z}\right)^{2}-a^{2}w^{2} = 1		 

(diff(w, z))^(2)- (a)^(2)* (w)^(2) = 1

(D[w, z])^(2)- (a)^(2)* (w)^(2) == 1
Failure Failure
Failed [300 / 300]
Result: -2.125000001-1.948557159*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -2.125000001-1.948557159*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: -2.125000001-1.948557159*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}


Result: -2.125000001-1.948557159*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-2.125, -1.9485571585149868]
Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.125, -1.9485571585149868]
Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.34.E9 ( d w d z ) 2 - a 2 ⁢ w 2 = - 1 superscript derivative 𝑤 𝑧 2 superscript 𝑎 2 superscript 𝑤 2 1 {\displaystyle{\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a% ^{2}w^{2}=-1}}
\left(\deriv{w}{z}\right)^{2}-a^{2}w^{2} = -1		 

(diff(w, z))^(2)- (a)^(2)* (w)^(2) = - 1

(D[w, z])^(2)- (a)^(2)* (w)^(2) == - 1
Failure Failure
Failed [272 / 300]
Result: -.125000001-1.948557159*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -.125000001-1.948557159*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: -.125000001-1.948557159*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}


Result: -.125000001-1.948557159*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [272 / 300]
Result: Complex[-0.12500000000000022, -1.9485571585149868]
Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.12500000000000022, -1.9485571585149868]
Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.34.E10 d w d z + a 2 ⁢ w 2 = 1 derivative 𝑤 𝑧 superscript 𝑎 2 superscript 𝑤 2 1 {\displaystyle{\displaystyle\frac{\mathrm{d}w}{\mathrm{d}z}+a^{2}w^{2}=1}}
\deriv{w}{z}+a^{2}w^{2} = 1		 

diff(w, z)+ (a)^(2)* (w)^(2) = 1

D[w, z]+ (a)^(2)* (w)^(2) == 1
Failure Failure
Failed [272 / 300]
Result: .125000001+1.948557159*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: .125000001+1.948557159*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: .125000001+1.948557159*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}


Result: .125000001+1.948557159*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [272 / 300]
Result: Complex[0.12500000000000022, 1.9485571585149868]
Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.12500000000000022, 1.9485571585149868]
Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.34.E11 w = A ⁢ cosh ⁡ ( a ⁢ z ) + B ⁢ sinh ⁡ ( a ⁢ z ) 𝑤 𝐴 𝑎 𝑧 𝐵 𝑎 𝑧 {\displaystyle{\displaystyle w=A\cosh\left(az\right)+B\sinh\left(az\right)}}
w = A\cosh@{az}+B\sinh@{az}		 

w = A*cosh(a*z)+ B*sinh(a*z)

w == A*Cosh[a*z]+ B*Sinh[a*z]
Failure Failure
Failed [300 / 300]
Result: .6001928989+.561234643*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -.6457530113+1.981963256*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: .9837329493+.425340516e-1*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}


Result: -.2074648399-3.005064943*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.6001928983405861, 0.5612346426489729]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.645753012062901, 1.9819632558589868]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.34.E12 w = ( 1 / a ) ⁢ sinh ⁡ ( a ⁢ z + c ) 𝑤 1 𝑎 𝑎 𝑧 𝑐 {\displaystyle{\displaystyle w=(1/a)\sinh\left(az+c\right)}}
w = (1/a)\sinh@{az+c}		 

w = (1/a)*sinh(a*z + c)

w == (1/a)*Sinh[a*z + c]
Failure Failure
Failed [300 / 300]
Result: -3.126061208-3.246674013*I
Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: .7188715257-.3314459800*I
Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: .265391293e-1+3.580357057*I
Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}


Result: .7673365303+.9636329126*I
Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-3.126061206522873, -3.246674011194613]
Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.7188715253469982, -0.33144598009263954]
Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.34.E13 w = ( 1 / a ) ⁢ cosh ⁡ ( a ⁢ z + c ) 𝑤 1 𝑎 𝑎 𝑧 𝑐 {\displaystyle{\displaystyle w=(1/a)\cosh\left(az+c\right)}}
w = (1/a)\cosh@{az+c}		 

w = (1/a)*cosh(a*z + c)

w == (1/a)*Cosh[a*z + c]
Failure Failure
Failed [300 / 300]
Result: 4.887803259+4.219013756*I
Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: 1.097709449+1.028092043*I
Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: 1.724372908-2.512669644*I
Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}


Result: 1.363701096+.4080617947*I
Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[4.887803257491119, 4.219013753952423]
Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.0977094487385304, 1.0280920432224616]
Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.34.E14 w = ( 1 / a ) ⁢ coth ⁡ ( a ⁢ z + c ) 𝑤 1 𝑎 hyperbolic-cotangent 𝑎 𝑧 𝑐 {\displaystyle{\displaystyle w=(1/a)\coth\left(az+c\right)}}
w = (1/a)\coth@{az+c}		 

w = (1/a)*coth(a*z + c)

w == (1/a)*Coth[a*z + c]
Failure Failure
Failed [300 / 300]
Result: .1990274306+.5049301211*I
Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: .4235738270+.6074604561*I
Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: .2119596261+.4924838498*I
Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}


Result: .5938323036-.1576784256*I
Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.19902743024251868, 0.504930121080845]
Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.423573826800421, 0.6074604562830159]
Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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