Derivative of 50*sin(y*cos(y))^(3)

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

   1. Let $u = \sin{\left(y \cos{\left(y \right)} \right)}$.

   2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d y} \sin{\left(y \cos{\left(y \right)} \right)}$:

      1. Let $u = y \cos{\left(y \right)}$.

      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 $\frac{d}{d y} y \cos{\left(y \right)}$:

         1. Apply the product rule:

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

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

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

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

            1. The derivative of cosine is negative sine:

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

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

         The result of the chain rule is:

         $$\left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \cos{\left(y \cos{\left(y \right)} \right)}$$

      The result of the chain rule is:

      $$3 \left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}$$

   So, the result is: $150 \left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}$

The answer is:

$$150 \left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}$$