Question

What is the derivative of `sqrt(x^2-1)`?

Answer 1

`dy/dx=x/sqrt{x^2-1}`

Explanation:

Let's equate the function to a variable `y`, so that
`y=sqrt{x^2-1}`

Now, I'll take another variable `t` and equate it as such,
`t=x^2-1`

So that makes the `y` function as `y=sqrtt`

Now, we are to find the derivative of `y` with respect to `x`. So that means we are to find `dy/dx`

Now, we can use chain rule to simplify our problem as
`dy/dx=dy/dt*dt/dx`

That makes it, `dy/dx=d/dt(sqrtt)*d/dx(x^2-1)`

Now, we know that for any Real value `n`, `d/dx(x^n)=nx^(n-1)`
, and that the derivative of a constant is zero. So
`dy/dx=1/{2sqrtt}*2x`

Now, `t` was taken as `t=x^2-1`, so substituting that back into the equation gives us the answer being searched for (after a few simplifications of course).