The chain rule applications
      Derivatives of the hyperbolic functions
         Derivatives of inverse hyperbolic functions
      Derivative of the inverse function
         Derivatives of the inverse trigonometric functions
Derivatives of the hyperbolic functions
We use the derivative of the exponential function and the chain rule to determine the derivative of the hyperbolic sine and the hyperbolic cosine functions.
We find derivative of the hyperbolic tangent and the hyperbolic cotangent functions applying the quotient rule.
Therefore, derivatives of the hyperbolic functions are
   
Derivatives of inverse hyperbolic functions
We use the derivative of the logarithmic function and the chain rule to find the derivative of inverse hyperbolic functions.
We use the same method to find derivatives of other inverse hyperbolic functions, thus
   
Derivative of the inverse function
If given a function  y = f (x)  the derivative of which  y' (x)  is not 0  then, the derivative of the inverse function  xf -1 (y)  is
Example:   Find the derivative  x'(y)  if the given function  f (x) = x + ln x.
Solution:   
Derivative of the inverse trigonometric functions
1)  The derivative of the inverse of the sine function  y = sin -1x,  | x | < 1  and  -p/2 < y < p/2  if  x = sin y, then 
2)  The derivative of the inverse of the cosine function  y = cos -1x = p/2 - sin -1x,  | x | < 1, 0 < y < p 
3)  The derivative of the inverse of the tangent function  y = tan -1x,  - oo  < x < oo   and  -p/2 < y < p/2 if  x = tan y, then 
4)  The derivative of the inverse of the cotangent function  y = cot -1x = p/2 - tan -1x,
5)  The derivative of the inverse of the secant function  y = sec -1x = cos -1(1/x),
6)  The derivative of the inverse of the cosecant function  y = csc -1x = sin -1(1/x),
Therefore, derivatives of the inverse trigonometric functions are
   
Calculus contents C
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