# §4.20 Derivatives and Differential Equations

 4.20.1 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\sin\/}\nolimits z$ $\displaystyle=\mathop{\cos\/}\nolimits z,$ 4.20.2 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\cos\/}\nolimits z$ $\displaystyle=-\mathop{\sin\/}\nolimits z,$ 4.20.3 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\tan\/}\nolimits z$ $\displaystyle={\mathop{\sec\/}\nolimits^{2}}z,$ 4.20.4 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\csc\/}\nolimits z$ $\displaystyle=-\mathop{\csc\/}\nolimits z\mathop{\cot\/}\nolimits z,$ 4.20.5 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\sec\/}\nolimits z$ $\displaystyle=\mathop{\sec\/}\nolimits z\mathop{\tan\/}\nolimits z,$ 4.20.6 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\cot\/}\nolimits z$ $\displaystyle=-{\mathop{\csc\/}\nolimits^{2}}z,$ 4.20.7 $\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\mathop{\sin\/}\nolimits z$ $\displaystyle=\mathop{\sin\/}\nolimits\!\left(z+\tfrac{1}{2}n\pi\right),$ 4.20.8 $\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\mathop{\cos\/}\nolimits z$ $\displaystyle=\mathop{\cos\/}\nolimits\!\left(z+\tfrac{1}{2}n\pi\right).$

With $a\neq 0$, the general solutions of the differential equations

 4.20.9 $\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+a^{2}w$ $\displaystyle=0,$ 4.20.10 $\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}+a^{2}w^{2}$ $\displaystyle=1,$ 4.20.11 $\displaystyle\frac{\mathrm{d}w}{\mathrm{d}z}-a^{2}w^{2}$ $\displaystyle=1,$

are respectively

 4.20.12 $\displaystyle w$ $\displaystyle=A\mathop{\cos\/}\nolimits\!\left(az\right)+B\mathop{\sin\/}% \nolimits\!\left(az\right),$ 4.20.13 $\displaystyle w$ $\displaystyle=(1/a)\mathop{\sin\/}\nolimits\!\left(az+c\right),$ Symbols: $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $a$: real or complex constant, $z$: complex variable and $c$: arbitrary constant Permalink: http://dlmf.nist.gov/4.20.E13 Encodings: TeX, pMML, png See also: Annotations for 4.20 4.20.14 $\displaystyle w$ $\displaystyle=(1/a)\mathop{\tan\/}\nolimits\!\left(az+c\right),$ Symbols: $\mathop{\tan\/}\nolimits\NVar{z}$: tangent function, $a$: real or complex constant, $z$: complex variable and $c$: arbitrary constant Permalink: http://dlmf.nist.gov/4.20.E14 Encodings: TeX, pMML, png See also: Annotations for 4.20

where $A,B,c$ are arbitrary constants.

For other differential equations see Kamke (1977, pp. 355–358 and 396–400).