Derivative of 11^(x^2+5x+14)

1. Let $u = \left(x^{2} + 5 x\right) + 14$.

2. $\frac{d}{d u} 11^{u} = 11^{u} \log{\left(11 \right)}$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{2} + 5 x\right) + 14\right)$:

   1. Differentiate $\left(x^{2} + 5 x\right) + 14$ term by term:

      1. Differentiate $x^{2} + 5 x$ term by term:

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

         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: $5$

         The result is: $2 x + 5$

      2. The derivative of the constant $14$ is zero.

      The result is: $2 x + 5$

   The result of the chain rule is:

   $$11^{\left(x^{2} + 5 x\right) + 14} \left(2 x + 5\right) \log{\left(11 \right)}$$

4. Now simplify:

   $$11^{x \left(x + 5\right) + 14} \left(2 x + 5\right) \log{\left(11 \right)}$$

The answer is:

$$11^{x \left(x + 5\right) + 14} \left(2 x + 5\right) \log{\left(11 \right)}$$