Derivative of sqrt(14x^2+16x+23)

1. Let $u = 14 x^{2} + 16 x + 23$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(14 x^{2} + 16 x + 23\right)$:

   1. Differentiate $14 x^{2} + 16 x + 23$ term by term:

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

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

         So, the result is: $28 x$

      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: $16$

      3. The derivative of the constant $23$ is zero.

      The result is: $28 x + 16$

   The result of the chain rule is:

   $$\frac{28 x + 16}{2 \sqrt{14 x^{2} + 16 x + 23}}$$

4. Now simplify:

   $$\frac{2 \cdot \left(7 x + 4\right)}{\sqrt{14 x^{2} + 16 x + 23}}$$

The answer is: