Hyperbolic functions

f(x) = xcosh1(x4) (x216)\displaystyle f(x) \ = \ xcosh^{-1}\bigg(\frac{x}{4}\bigg) \ -\sqrt(x^{2}-16)

f  (x) = (1)cosh1(x4)+x1/4((x2/16)1)12(x216)1/2(2x)\displaystyle f \ ' \ (x) \ = \ (1)cosh^{-1}\bigg(\frac{x}{4}\bigg)+x \frac{1/4}{\sqrt((x^{2}/16)-1)} -\frac{1}{2}(x^{2}-16)^{-1/2}(2x)

\(\displaystyle = \ cosh^-1}\bigg(\frac{x}{4}\bigg)+\frac{x}{\sqrt(x^{2}-16)}-\frac{x}{\sqrt(x^{2}-16)}\)

= cosh1(x4)\displaystyle = \ cosh^{-1}\bigg(\frac{x}{4}\bigg)

Hence f  (6) = cosh1(32) = ln(3+52). QED\displaystyle Hence \ f \ ' \ (6) \ = \ cosh^{-1}\bigg(\frac{3}{2}\bigg) \ = \ ln\bigg(\frac{3+\sqrt 5}{2}\bigg). \ QED
 
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