In order to find h'(x) we need to know a few things:

The derivative of log(f(x)), and of sqrt(x^2-1).

We need to know the Chain rule. Given h(x)=log(x+sqrt(x^2-1)), we start by differentiating log(f(x)), where f(x) = x+sqrt(x^2-1):

d/dx (log(f(x))) = **1/f(x) * d/dx(f(x)).**

Now find the derivative of f(x), that is

d/dx (x+sqrt(g(x))), where g(x) = x^2-1

We have d/dx(x+sqrt(g(x))) = **1 + 1/(2*sqrt(g(x)) * d/dx(g(x))**

Finally we are left with finding the derivative of g(x), simple enough.

d/dx (x^2-1) = **2*x**

Back substituting we have

d/dx (h(x)) = 1/(x+sqrt(x^2-1)) * (1 + 1/(2*sqrt(x^2-1))*2*x)

Simplifying, h'(x) = 1/(x+sqrt(x^2-1)) * (1 + x/(sqrt(x^2-1))

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