Derivative of 1/sqrt(1-x)

1. Let $u = \sqrt{1 - x}$.

2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{1 - x}$:

   1. Let $u = 1 - x$.

   2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - x\right)$:

      1. Differentiate $1 - x$ term by term:

         1. The derivative of the constant $1$ is zero.

         2. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: $x$ goes to $1$

            So, the result is: $-1$

         The result is: $-1$

      The result of the chain rule is:

      $$- \frac{1}{2 \sqrt{1 - x}}$$

   The result of the chain rule is:

   $$\frac{1}{2 \left(1 - x\right)^{\frac{3}{2}}}$$

The answer is: