Derivative of x+10*sin(2*pi*x/12)

1. Differentiate $x + 10 \sin{\left(\frac{2 \pi x}{12} \right)}$ term by term:

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

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

      1. Let $u = \frac{2 \pi x}{12}$.

      2. The derivative of sine is cosine:

         $$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$$

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 \pi x}{12}$:

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

            1. 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: $2 \pi$

            So, the result is: $\frac{\pi}{6}$

         The result of the chain rule is:

         $$\frac{\pi \cos{\left(\frac{2 \pi x}{12} \right)}}{6}$$

      So, the result is: $\frac{5 \pi \cos{\left(\frac{2 \pi x}{12} \right)}}{3}$

   The result is: $\frac{5 \pi \cos{\left(\frac{2 \pi x}{12} \right)}}{3} + 1$

2. Now simplify:

   $$\frac{5 \pi \cos{\left(\frac{\pi x}{6} \right)}}{3} + 1$$

The answer is:

$$\frac{5 \pi \cos{\left(\frac{\pi x}{6} \right)}}{3} + 1$$