Derivative of y=sinx+1/3sin3x

1. Differentiate $\sin{\left(x \right)} + \frac{\sin{\left(3 x \right)}}{3}$ term by term:

   1. The derivative of sine is cosine:

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

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

      1. Let $u = 3 x$.

      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 $\frac{d}{d x} 3 x$:

         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: $3$

         The result of the chain rule is:

         $$3 \cos{\left(3 x \right)}$$

      So, the result is: $\cos{\left(3 x \right)}$

   The result is: $\cos{\left(x \right)} + \cos{\left(3 x \right)}$

The answer is:

$$\cos{\left(x \right)} + \cos{\left(3 x \right)}$$