Derivatives of Inverse Hyperbolic Functions

Inverse trigonometric functions are often found in physical applications.

Derivatives of Inverse Hyperbolic Functions

1. Derivative of Inverse Hyperbolic Sine Function

   

   y is the value lying between -∞ and ∞ and x is the value of sinh y from -∞ to ∞. The slope of the curve is always positive, imply dy/dx is always positive.

   

   Proof:

   

2. Derivative of Inverse Hyperbolic Cosine Function

   

   y is the value lying between 0 and ∞  or between 0 and -∞ and x is the value of sinh y from 1 to ∞. The slope of the curve is either always positive or always negative, imply dy/dx is either always positive or always negative.

   or

   Proof:

   

3. Derivative of Inverse Hyperbolic Tangent Function

   

   y is the value lying between -∞ and ∞ and x is the value of sinh y from -1 to 1. The slope of the curve is always positive, imply dy/dx is always positive.

   

   Proof:

   

    

4. Derivative of Inverse Hyperbolic Cotangent Function

   

   y is the value lying between -∞ and ∞ and x is the value of sinh y from -1 to ∞ and 1 to ∞. The slope of the curve is always negative, imply dy/dx is always negative.

   

   Proof:

   

5. Derivative of Inverse Hyperbolic Secant Function

   

   y is the value lying between 0 and ∞  or between 0 and -∞ and x is the value of sinh y from 0 to 1. The slope of the curve is either always positive or always negative, imply dy/dx is either always positive or always negative.

     or

   Proof:

   

6. Derivative of Inverse Hyperbolic Cosecant Function

   

   y is the value lying between -∞ and ∞ and x is the value of sinh y from -∞ to -1  and 1 to ∞. The slope of the curve is always negative, imply dy/dx is always negative.

   

   Proof: