Derivative of y=3cos5x+2ctg(x2)

1. Differentiate $3 \cos{\left(5 x \right)} + 2 \cot{\left(x_{2} \right)}$ term by term:

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

      1. Let $u = 5 x$.

      2. The derivative of cosine is negative sine:

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

      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:

         $$- 5 \sin{\left(5 x \right)}$$

      So, the result is: $- 15 \sin{\left(5 x \right)}$

   2. The derivative of the constant $2 \cot{\left(x_{2} \right)}$ is zero.

   The result is: $- 15 \sin{\left(5 x \right)}$

The answer is:

$$- 15 \sin{\left(5 x \right)}$$