Derivative of 7*cos(x)+12x^5

1. Differentiate $12 x^{5} + 7 \cos{\left(x \right)}$ term by term:

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

      1. The derivative of cosine is negative sine:

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

      So, the result is: $- 7 \sin{\left(x \right)}$

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

      1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$

      So, the result is: $60 x^{4}$

   The result is: $60 x^{4} - 7 \sin{\left(x \right)}$

The answer is:

$$60 x^{4} - 7 \sin{\left(x \right)}$$