Question

How do you find the derivative of `y = tanh^-1 (1/x)`?

Answer 1

`dy/dx=1/(1-x^2)`

Explanation:

`y=tanh^-1(1/x) => tanhy=1/x`

Differentiating implicitly gives:

`(1-tanh^2y)dy/dx=-1/x^2`
`:. (1-(1/x)^2)dy/dx=-1/x^2`
`:. dy/dx=-1/((x^2)(1-(1/x)^2))`
`:. dy/dx=-1/((x^2)(1-1/x^2))`
`:. dy/dx=-1/(x^2-1)`
`:. dy/dx=1/(1-x^2)`