Derivative of x=3costy=3sint

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

   1. Let $u = t y$.

   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 $\frac{\partial}{\partial y} t y$:

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

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

         So, the result is: $t$

      The result of the chain rule is:

      $$- t \sin{\left(t y \right)}$$

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

The answer is:

$$- 3 t \sin{\left(t y \right)}$$