The Derivative of Hyperbolic and Area Functions (with examples)

• Example 1 Determine the first derivative of a function \(y = x \sinh x\)
  Solution \begin{eqnarray}y' &=& \sinh x + x\cosh x\end{eqnarray}
• Example2 Determine the first derivative of a function \(y = \frac{x^2}{\cosh x}\)
  Solution \begin{eqnarray}y' &=& \frac{2x\cosh x - x^2\cdot \cosh x}{\cosh^2 x}\end{eqnarray}
• Example 3 Determine the first dderivative of a function \(y = \tanh x - x \)
  Solution \begin{eqnarray}y' &=& \frac{1}{\cosh^2 x} -1 y' &=& \frac{1 - \cosh^2 x}{\cosh^2 x}\end{eqnarray}
• Example 4 Determine the first derivative of a function \(y = \frac{\coth x}{\ln x}\)
  Solution \begin{eqnarray}y' &=& \frac{-3\frac{1}{\sinh^2 x}\ln x - 3 \coth x \frac{1}{x}}{\ln^2 x} \\\nonumber y'&=& -3 \frac{\frac{\ln x}{\sinh^2 x}+ \frac{\coth x}{x}}{\ln^2 x}\end{eqnarray}
• Example 5 Determine the first derivative of a function \(y = arctan x - \mathrm{arctanh} x\)
  Solution \begin{eqnarray}y' &=& y = arctan x - \mathrm{arctanh} x \\\nonumber y' &=& \frac{1}{1+x^2} - \frac{1}{1-x^2} \\\nonumber y'&=& \frac{1-x^2 - 1- x^2 }{(1-x^2)(1+x^2)}\\\nonumber y' &=& \frac{-2x^2}{(1-x^2)(1+x^2)}\end{eqnarray}
• Example 6 Determine the first derivative of a function \(y = \arcsin x \mathrm{arcsinh} x\)
  Solution \begin{eqnarray}y = \arcsin x \mathrm{arcsinh} x \\\nonumber y' &=& \frac{1}{\sqrt{1-x^2}} \mathrm{arcsinh} x + \arcsin x \frac{1}{\sqrt{1+x^2}}\end{eqnarray}
• Example 7 Determine the first derivative of the function \(y = \frac{\mathrm{arccosh}x}{x} \)
  Solution \begin{eqnarray}y &=& \frac{\mathrm{arccosh}x}{x} \\\nonumber y' &=& \frac{\frac{x}{\sqrt{x^2-1}} - \mathrm{arccosh}x}{x^2} \end{eqnarray}
• Example 8 Determine the first derivative of a function \(y = \frac{\arctan x}{1-x^2}\)
  Solution \begin{eqnarray}y'&=& \frac{\frac{1}{1-x^2}(1-x^2) - \arctan x (-2x)}{(1-x^2)^2} \\\nonumber y'&=& \frac{1+2x\arctan x}{(1-x^2)^2}\end{eqnarray}