text: §4.34 Derivatives and Differential Equations
  
   4.34.1 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sinh z$ $\displaystyle=\cosh z,$ 4.34.2 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cosh z$ $\displaystyle=\sinh z,$ 4.34.3 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\tanh z$ $\displaystyle={\operatorname{sech}}^{2}z,$ 4.34.4 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{csch}z$ $\displaystyle=-\operatorname{csch}z\coth z,$ 4.34.5 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sech}z$ $\displaystyle=-\operatorname{sech}z\tanh z,$ 4.34.6 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\coth z$ $\displaystyle=-{\operatorname{csch}}^{2}z.$
  
  With $a\neq 0$, the general solutions of the differential equations
  
   4.34.7 $\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-a^{2}w$ $\displaystyle=0,$ 4.34.8 $\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}$ $\displaystyle=1,$ 4.34.9 $\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}$ $\displaystyle=-1,$ 4.34.10 $\displaystyle\frac{\mathrm{d}w}{\mathrm{d}z}+a^{2}w^{2}$ $\displaystyle=1,$
  
  are respectively
  
   4.34.11 $\displaystyle w$ $\displaystyle=A\cosh\left(az\right)+B\sinh\left(az\right),$ 4.34.12 $\displaystyle w$ $\displaystyle=(1/a)\sinh\left(az+c\right),$ 4.34.13 $\displaystyle w$ $\displaystyle=(1/a)\cosh\left(az+c\right),$ 4.34.14 $\displaystyle w$ $\displaystyle=(1/a)\coth\left(az+c\right),$
  
  where $A,B,c$ are arbitrary constants.
  
  For other differential equations see Kamke (1977, pp. 289–400).
explanation: The document is a section from a mathematical text, specifically dealing with derivatives and differential equations involving hyperbolic functions. It lists the derivatives of hyperbolic functions and provides the general solutions to several differential equations. The text does not contain a specific question or a question-and-answer format; it is purely expository.
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distilabel_metadata.content: §4.34 Derivatives and Differential Equations
  
   4.34.1 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sinh z$ $\displaystyle=\cosh z,$ 4.34.2 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cosh z$ $\displaystyle=\sinh z,$ 4.34.3 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\tanh z$ $\displaystyle={\operatorname{sech}}^{2}z,$ 4.34.4 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{csch}z$ $\displaystyle=-\operatorname{csch}z\coth z,$ 4.34.5 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sech}z$ $\displaystyle=-\operatorname{sech}z\tanh z,$ 4.34.6 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\coth z$ $\displaystyle=-{\operatorname{csch}}^{2}z.$
  
  With $a\neq 0$, the general solutions of the differential equations
  
   4.34.7 $\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-a^{2}w$ $\displaystyle=0,$ 4.34.8 $\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}$ $\displaystyle=1,$ 4.34.9 $\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}$ $\displaystyle=-1,$ 4.34.10 $\displaystyle\frac{\mathrm{d}w}{\mathrm{d}z}+a^{2}w^{2}$ $\displaystyle=1,$
  
  are respectively
  
   4.34.11 $\displaystyle w$ $\displaystyle=A\cosh\left(az\right)+B\sinh\left(az\right),$ 4.34.12 $\displaystyle w$ $\displaystyle=(1/a)\sinh\left(az+c\right),$ 4.34.13 $\displaystyle w$ $\displaystyle=(1/a)\cosh\left(az+c\right),$ 4.34.14 $\displaystyle w$ $\displaystyle=(1/a)\coth\left(az+c\right),$
  
  where $A,B,c$ are arbitrary constants.
  
  For other differential equations see Kamke (1977, pp. 289–400).
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distilabel_metadata.raw_output_q_a_detector_0: The document is a section from a mathematical text, specifically dealing with derivatives and differential equations involving hyperbolic functions. It lists the derivatives of hyperbolic functions and provides the general solutions to several differential equations. The text does not contain a specific question or a question-and-answer format; it is purely expository.
  
  Label: no-question