Derivative of y=x^2ln(e^sin(3x))

1. Apply the product rule:

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

   $f{\left(x \right)} = x^{2}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the power rule: $x^{2}$ goes to $2 x$

   $g{\left(x \right)} = \log{\left(e^{\sin{\left(3 x \right)}} \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = e^{\sin{\left(3 x \right)}}$.

   2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

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

      1. Let $u = \sin{\left(3 x \right)}$.

      2. The derivative of $e^{u}$ is itself.

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

         1. Let $u = 3 x$.

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

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

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

               So, the result is: $3$

            The result of the chain rule is:

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

         The result of the chain rule is:

         $$3 e^{\sin{\left(3 x \right)}} \cos{\left(3 x \right)}$$

      The result of the chain rule is:

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

   The result is: $3 x^{2} \cos{\left(3 x \right)} + 2 x \log{\left(e^{\sin{\left(3 x \right)}} \right)}$

2. Now simplify:

   $$x \left(3 x \cos{\left(3 x \right)} + 2 \sin{\left(3 x \right)}\right)$$

The answer is: