Derivative of 3cosx-12sinx5x

1. Differentiate $- x 5 \cdot 12 \sin{\left(x \right)} + 3 \cos{\left(x \right)}$ 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: $- 3 \sin{\left(x \right)}$

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

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

         1. Apply the product rule:

            $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

            $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

            1. Apply the power rule: $x$ goes to $1$

            $g{\left(x \right)} = \sin{\left(x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

            1. The derivative of sine is cosine:

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

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

         So, the result is: $60 x \cos{\left(x \right)} + 60 \sin{\left(x \right)}$

      So, the result is: $- 60 x \cos{\left(x \right)} - 60 \sin{\left(x \right)}$

   The result is: $- 60 x \cos{\left(x \right)} - 63 \sin{\left(x \right)}$

The answer is:

$$- 60 x \cos{\left(x \right)} - 63 \sin{\left(x \right)}$$