Derivative of 3*sin(t)-sin(3t)

1. Differentiate $3 \sin{\left(t \right)} - \sin{\left(3 t \right)}$ 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 sine is cosine:

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

      So, the result is: $3 \cos{\left(t \right)}$

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

      1. Let $u = 3 t$.

      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 $\frac{d}{d t} 3 t$:

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

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

            So, the result is: $3$

         The result of the chain rule is:

         $$3 \cos{\left(3 t \right)}$$

      So, the result is: $- 3 \cos{\left(3 t \right)}$

   The result is: $3 \cos{\left(t \right)} - 3 \cos{\left(3 t \right)}$

The answer is:

$$3 \cos{\left(t \right)} - 3 \cos{\left(3 t \right)}$$