Elementary Functions - 4.20 Derivatives and Differential Equations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.20.E1 d d z ⁡ sin ⁡ z = cos ⁡ z derivative 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sin z=\cos z}}
\deriv{}{z}\sin@@{z} = \cos@@{z}		 

diff(sin(z), z) = cos(z)

D[Sin[z], z] == Cos[z]
Successful Successful - Successful [Tested: 7]
4.20.E2 d d z ⁡ cos ⁡ z = - sin ⁡ z derivative 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cos z=-\sin z}}
\deriv{}{z}\cos@@{z} = -\sin@@{z}		 

diff(cos(z), z) = - sin(z)

D[Cos[z], z] == - Sin[z]
Successful Successful - Successful [Tested: 7]
4.20.E3 d d z ⁡ tan ⁡ z = sec 2 ⁡ z derivative 𝑧 𝑧 2 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\tan z={\sec^{2}}z}}
\deriv{}{z}\tan@@{z} = \sec^{2}@@{z}		 

diff(tan(z), z) = (sec(z))^(2)

D[Tan[z], z] == (Sec[z])^(2)
Successful Successful - Successful [Tested: 7]
4.20.E4 d d z ⁡ csc ⁡ z = - csc ⁡ z ⁢ cot ⁡ z derivative 𝑧 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\csc z=-\csc z\cot z}}
\deriv{}{z}\csc@@{z} = -\csc@@{z}\cot@@{z}		 

diff(csc(z), z) = - csc(z)*cot(z)

D[Csc[z], z] == - Csc[z]*Cot[z]
Successful Successful - Successful [Tested: 7]
4.20.E5 d d z ⁡ sec ⁡ z = sec ⁡ z ⁢ tan ⁡ z derivative 𝑧 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sec z=\sec z\tan z}}
\deriv{}{z}\sec@@{z} = \sec@@{z}\tan@@{z}		 

diff(sec(z), z) = sec(z)*tan(z)

D[Sec[z], z] == Sec[z]*Tan[z]
Successful Successful - Successful [Tested: 7]
4.20.E6 d d z ⁡ cot ⁡ z = - csc 2 ⁡ z derivative 𝑧 𝑧 2 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cot z=-{\csc^{2}}z}}
\deriv{}{z}\cot@@{z} = -\csc^{2}@@{z}		 

diff(cot(z), z) = - (csc(z))^(2)

D[Cot[z], z] == - (Csc[z])^(2)
Successful Successful - Successful [Tested: 7]
4.20.E7 d n d z n ⁡ sin ⁡ z = sin ⁡ ( z + 1 2 ⁢ n ⁢ π ) derivative 𝑧 𝑛 𝑧 𝑧 1 2 𝑛 𝜋 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\sin z=% \sin\left(z+\tfrac{1}{2}n\pi\right)}}
\deriv[n]{}{z}\sin@@{z} = \sin@{z+\tfrac{1}{2}n\pi}		 

diff(sin(z), [z$(n)]) = sin(z +(1)/(2)*n*Pi)

D[Sin[z], {z, n}] == Sin[z +Divide[1,2]*n*Pi]
Successful Successful - Successful [Tested: 21]
4.20.E8 d n d z n ⁡ cos ⁡ z = cos ⁡ ( z + 1 2 ⁢ n ⁢ π ) derivative 𝑧 𝑛 𝑧 𝑧 1 2 𝑛 𝜋 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\cos z=% \cos\left(z+\tfrac{1}{2}n\pi\right)}}
\deriv[n]{}{z}\cos@@{z} = \cos@{z+\tfrac{1}{2}n\pi}		 

diff(cos(z), [z$(n)]) = cos(z +(1)/(2)*n*Pi)

D[Cos[z], {z, n}] == Cos[z +Divide[1,2]*n*Pi]
Successful Successful - Successful [Tested: 21]
4.20.E9 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
4.20.E10 ( 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
4.20.E11 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 [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
4.20.E12 w = A ⁢ cos ⁡ ( a ⁢ z ) + B ⁢ sin ⁡ ( a ⁢ z ) 𝑤 𝐴 𝑎 𝑧 𝐵 𝑎 𝑧 {\displaystyle{\displaystyle w=A\cos\left(az\right)+B\sin\left(az\right)}}
w = A\cos@{az}+B\sin@{az}		 

w = A*cos(a*z)+ B*sin(a*z)

w == A*Cos[a*z]+ B*Sin[a*z]
Failure Failure
Failed [300 / 300]
Result: 1.138704571+1.826991634*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: -1.586785764-.8180862806*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: 1.979513822-1.625744019*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: -.8007246334+.1975056737*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[1.138704570618858, 1.8269916342928783]
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[-1.5867857625486925, -0.8180862808059206]
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
4.20.E13 w = ( 1 / a ) ⁢ sin ⁡ ( a ⁢ z + c ) 𝑤 1 𝑎 𝑎 𝑧 𝑐 {\displaystyle{\displaystyle w=(1/a)\sin\left(az+c\right)}}
w = (1/a)\sin@{az+c}		 

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

w == (1/a)*Sin[a*z + c]
Failure Failure
Failed [300 / 300]
Result: .5761075690+1.016359912*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: -.288669860e-1-.3275339707*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: -.1554713530-.2104590960*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: .6937358929+1.037178419*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[0.5761075684969701, 1.0163599120046827]
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[-0.028866985825810376, -0.3275339701177746]
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
4.20.E14 w = ( 1 / a ) ⁢ tan ⁡ ( a ⁢ z + c ) 𝑤 1 𝑎 𝑎 𝑧 𝑐 {\displaystyle{\displaystyle w=(1/a)\tan\left(az+c\right)}}
w = (1/a)\tan@{az+c}		 

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

w == (1/a)*Tan[a*z + c]
Failure Failure
Failed [300 / 300]
Result: 1.000937702+.460093509e-1*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: .7686167751-.1524919258*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: .9655903492+1.180557377*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: .7863384613+.9337431086*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[1.0009377022129278, 0.04600935086169866]
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[0.7686167748870922, -0.1524919257161706]
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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