Hyperbolic Trigonometric Functions

Logarithmic form for sinh^-1(x)

Let

Then

x = sinh(y),

so

Clearing fractions and moving everything to one side , we get

and multiplying by e ^y , we get

This is a quadratic equation in e ^y , so we may solve it by the quadratic formula :

Since e^y must be positive , we can discard the negative square root here to get

so,

Logarithmic form for cosh^-1(x) (x ≥ 1)

Let y = cosh^-1(x).

Then

x = cosh(y),

so

Clearing fractions and moving everything to one side, we get

and multiplying by e^y , we get

This is a quadratic equation in e^y , so we may solve it by the quadratic formula:

Since e^y must be positive, we can discard the negative square root here to get

so

Logarithmic form for tanh^-1(x)

Let

y = tanh^-1(x).

Then

x = tanh(y),

so

Clearing fractions and moving everything to one side, we get

So,

Logarithmic form for coth^-1(x)

Let

y = coth^-1(x).

Then

x = coth(y),

so

Clearing fractions and moving everything to one side, we get

So,

Logarithmic form for sech^-1(x)

Let

y = sech^-1(x).

Then

x = sech(y),

so

Clearing fractions and moving everything to one side, we get

This is a quadratic equation in e^y , so we can solve this using the quadratic formula:

Since e^y must be positive, we must have

Taking the natural logarithm, we get

Logarithmic form for csch^-1(x)
 

Let

y = csch^-1(x).

Then

x = csch(y),

so

Clearing fractions and moving everything to one side, we get

This is a quadratic equation in e^y , so we can solve this using the quadratic formula:

Since e^y must be positive, we must have

We can write these two equations as one as follows:

Taking the natural logarithm, we get

Derivatives of Inverse Hyperbolic Functions
Verify the fol lowing derivatives :