How do you find the derivative of the function #y=cosh^-1((sqrt(x))#?

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Answer 1 · 2016-12-30T12:57:53

The answer is #=1/(2sqrtxsqrt(x-1))#

We need

#(sqrtx)'=1/(2sqrtx)#

#(coshx)'=sinhx#

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

Here, we have

#y=cosh^(-1)(sqrtx)#

Therefore,

#coshy=sqrtx#

Taking the derivatives on both sides

#(coshy)'=(sqrtx)'#

#sinhydy/dx=1/(2sqrtx)#

#dy/dx=1/(2sqrtxsinhy)#

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

#sinh^2y=cos^2y-1#

#sinh^2y=x-1#

#sinhy=sqrt(x-1)#

Therefore,

#dy/dx=1/(2sqrtxsqrt(x-1))#