How do you differentiate $y=cosh^-1sqrt(x^2+1)$ ?

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Answer 1 · 2017-04-12T09:44:23

The answer is $=x/(|x|sqrt(x^2+1)$

We need

$(coshx)'=sinhx$

$cosh^2x-sinh^2x=1$

$sinh^2x=cosh^2x-1$

Let rewrite the equation

$y=cosh^-1(sqrt(x^2+1))$

So,

$coshy=sqrt(x^2+1)$

Differentiating both sides

$(coshy)'=(sqrt(x^2+1))'$

$sinhy*dy/dx=(2x)/(2sqrt(x^2+1))=x/sqrt(x^2+1)$

$dy/dx=1/sinhy*x/sqrt(x^2+1)$

We calculate $sinhy$

$sinh^2y=cosh^2y-1$

$=x^2+1-1=x^2$

$sinhy=|x|$

$dy/dx=x/(|x|sqrt(x^2+1)$

How do you differentiate y=cosh^-1sqrt(x^2+1) y=cosh^-1sqrt(x^2+1)?