Question

Derivatives of. 1/undroot x square - 1

Answer 1

The derivative of $\frac{1}{\sqrt{x^2 - 1}}$ is $-\frac{x}{(x^2 - 1)^{3/2}}$.

Answer 2

The problem asks for the derivative of the function $y = \frac{1}{\sqrt{x^2 - 1}}$.

Explanation:

1. Rewrite the function: The given function can be written in exponential form: $y = (x^2 - 1)^{-1/2}$

2. Apply the Chain Rule: Let $u = x^2 - 1$. Then $y = u^{-1/2}$. The chain rule states $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

   • Find $\frac{dy}{du}$: $\frac{dy}{du} = \frac{d}{du}(u^{-1/2}) = -\frac{1}{2} u^{-1/2 - 1} = -\frac{1}{2} u^{-3/2}$
   • Find $\frac{du}{dx}$: $\frac{du}{dx} = \frac{d}{dx}(x^2 - 1) = 2x$

3. Combine the derivatives: $\frac{dy}{dx} = \left(-\frac{1}{2} u^{-3/2}\right) \cdot (2x)$

4. Substitute back $u = x^2 - 1$ and simplify: $\frac{dy}{dx} = -\frac{1}{2} (x^2 - 1)^{-3/2} \cdot (2x)$ $\frac{dy}{dx} = -x (x^2 - 1)^{-3/2}$

5. Express in radical form (optional): $\frac{dy}{dx} = -\frac{x}{(x^2 - 1)^{3/2}}$ or $\frac{dy}{dx} = -\frac{x}{\sqrt{(x^2 - 1)^3}}$