Question

How do you find the derivative of `sqrt(x^2-1)`?

Answer 1

Use the power rule and chain rule to get `x/(sqrt(x^2-1))`.

Explanation:

Recognize that `sqrt(x^2-1)` can be written as `(x^2-1)^(1/2)`. Now that we have it in this form, we can use the power rule to get:
`d/dx=1/2(x^2-1)^(-1/2)`

We also need to use the chain rule; the derivative of the "inside" function (`x^2`) is `2x`, so we multiply `1/2(x^2-1)^(-1/2)` by `2x`:
`d/dx=2x*1/2(x^2-1)^(-1/2)`

Canceling the `2`s and using the properties of exponents yields:
`d/dx=x/((x^2-1)^(1/2))`

Since the problem was given to us in radical form, convert back to get the final answer:
`d/dx=x/(sqrt(x^2-1))`