Derivative of 1/(2sqrtx^2-6x+11)

1. Let $u = \left(2 \left(\sqrt{x}\right)^{2} - 6 x\right) + 11$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(2 \left(\sqrt{x}\right)^{2} - 6 x\right) + 11\right)$:

   1. Differentiate $\left(2 \left(\sqrt{x}\right)^{2} - 6 x\right) + 11$ term by term:

      1. Differentiate $2 \left(\sqrt{x}\right)^{2} - 6 x$ term by term:

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

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

            2. Apply the power rule: $u^{2}$ goes to $2 u$

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

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

               The result of the chain rule is:

               $$1$$

            So, the result is: $2$

         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: $-6$

         The result is: $-4$

      2. The derivative of the constant $11$ is zero.

      The result is: $-4$

   The result of the chain rule is:

   $$\frac{4}{\left(\left(2 \left(\sqrt{x}\right)^{2} - 6 x\right) + 11\right)^{2}}$$

4. Now simplify:

   $$\frac{4}{\left(11 - 4 x\right)^{2}}$$

The answer is:

$$\frac{4}{\left(11 - 4 x\right)^{2}}$$