Derivative of x*exp(x*(x/9))

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 $1$

   $g{\left(x \right)} = e^{x \frac{x}{9}}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = x \frac{x}{9}$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x \frac{x}{9}$:

      1. Apply the quotient rule, which is:

         $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

         $f{\left(x \right)} = x^{2}$ and $g{\left(x \right)} = 9$.

         To find $\frac{d}{d x} f{\left(x \right)}$:

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

         To find $\frac{d}{d x} g{\left(x \right)}$:

         1. The derivative of the constant $9$ is zero.

         Now plug in to the quotient rule:

         $\frac{2 x}{9}$

      The result of the chain rule is:

      $$\frac{2 x e^{x \frac{x}{9}}}{9}$$

   The result is: $\frac{2 x^{2} e^{x \frac{x}{9}}}{9} + e^{x \frac{x}{9}}$

2. Now simplify:

   $$\frac{\left(2 x^{2} + 9\right) e^{\frac{x^{2}}{9}}}{9}$$

The answer is:

$$\frac{\left(2 x^{2} + 9\right) e^{\frac{x^{2}}{9}}}{9}$$