Derivative of x*exp(3x)*cos3x

1. Apply the product rule:

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

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

   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 $1$

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

      1. Let $u = 3 x$.

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

      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:

         $$3 e^{3 x}$$

      The result is: $3 x e^{3 x} + e^{3 x}$

   $g{\left(x \right)} = \cos{\left(3 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 3 x$.

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

      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:

      $$- 3 \sin{\left(3 x \right)}$$

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

2. Now simplify:

   $$\left(- 3 x \sin{\left(3 x \right)} + \left(3 x + 1\right) \cos{\left(3 x \right)}\right) e^{3 x}$$

The answer is:

$$\left(- 3 x \sin{\left(3 x \right)} + \left(3 x + 1\right) \cos{\left(3 x \right)}\right) e^{3 x}$$