Derivative of y=sin5x+7x

1. Differentiate $7 x + \sin{\left(5 x \right)}$ term by term:

   1. Let $u = 5 x$.

   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 $\frac{d}{d x} 5 x$:

      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: $5$

      The result of the chain rule is:

      $$5 \cos{\left(5 x \right)}$$

   4. 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: $7$

   The result is: $5 \cos{\left(5 x \right)} + 7$

The answer is:

$$5 \cos{\left(5 x \right)} + 7$$