Here we have to find the Derivatives of inverse hyperbolic Functions. To calculate derivatives of inverse Hyperbolic Functions, first we must study what are Inverse Hyperbolic Functions. Inverse hyperbolic functions are represented by sinh - 1 ( x ), cosh - 1 ( x ), tanh - 1 ( x ), coth - 1 ( x ), sech - 1 ( x ), and cosech - 1 ( x ).
Suppose we have y = sin– 1h ( x ), it can be written as x = sinh ( y ).
Value of x = sinh ( y ) = ( e y – e - y ) / 2 . So we can find inverse by solving for ' y ' in terms of 'x'.
Now let us find derivative of y = sinh – 1 x .
From above concept x = sinh ( y ).
Therefore x = 1 / 2 ( e y – e - y ),
=> 2x = ( e y – e - y ),
=> 2x . e y = e2y – 1,
e y = 2x ∓√ ( 4x2 + 4 ) / 2 = x ∓√ ( x2 + 1 ),
ey will always be positive, so
ey = x + √ ( x2 + 1 ),
=> y = ln ( x + √ ( x2 + 1 ) ),
Therefore Sinh – 1 x = ln ( x + √ ( x2 + 1 ) ).
Similarly we can calculate derivatives of other inverse hyperbolic functions, the derivatives of inverse hyperbolic functions are shown below-
1) Sinh – 1 x = ln ( x + √ ( x2 + 1 ) ).
2) cosh-1 x = ln ( x + √ ( x2 - 1 ) ).
3) tanh-1 x = ( 1 / 2 )ln [ ( 1 + x ) / ( 1 – x ) ].