Let y-tan h -1 x so tan hy -xExpress tan hy in terms of ey and hence show that e 2y - 1-x 1-xDeduce that tan h -1 x - 1 2 ln 1-x 1-x Do not forget that tan h -1 x is only defined for - x - - 1

Given tanh^ 1x=y ⇒ tanhy=x ⇒ tanhy =e^2y 1e^2y+1 ⇒e^2y 1e^2y+1=x ⇒ e^2y=1+x1 x ⇒ y=12ln(1+x1 x) ⇒ y=tanh^ 1x=12ln(1+x1 x) ...

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First Principle of Differentiation

Let y=tan h...

Question

Let $y = tan h^{- 1} x$ so $tan h y = x$
Express $tan h y$ in terms of $e^{y}$ and hence show that
$e^{2 y} = \frac{1 + x}{1 - x}$
Deduce that $tan h^{- 1} x = \frac{1}{2} ln (\frac{1 + x}{1 - x})$
(Do not forget that $tan h^{- 1} x$ is only defined for $| x | < 1$ )

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