Derivative of y=(m+n)x+sinmx

1. Differentiate $x \left(m + n\right) + \sin{\left(m x \right)}$ 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: $x$ goes to $1$

      So, the result is: $m + n$

   2. Let $u = m x$.

   3. The derivative of sine is cosine:

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

   4. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} m x$:

      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: $m$

      The result of the chain rule is:

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

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

The answer is:

$$m \cos{\left(m x \right)} + m + n$$