Derivative of y'=sin(50x+p/3)

1. Let $u = \frac{p}{3} + 50 x$.

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 $\frac{\partial}{\partial x} \left(\frac{p}{3} + 50 x\right)$:

   1. Differentiate $\frac{p}{3} + 50 x$ 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: $50$

      2. The derivative of the constant $\frac{p}{3}$ is zero.

      The result is: $50$

   The result of the chain rule is:

   $$50 \cos{\left(\frac{p}{3} + 50 x \right)}$$

4. Now simplify:

   $$50 \cos{\left(\frac{p}{3} + 50 x \right)}$$

The answer is:

$$50 \cos{\left(\frac{p}{3} + 50 x \right)}$$