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)#