Derivative of x^2*lnx+e^2x

1. Differentiate $x^{2} \log{\left(x \right)} + x e^{2}$ term by term:

   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 $2 x$

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

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

      The result is: $2 x \log{\left(x \right)} + x$

   2. 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: $e^{2}$

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

2. Now simplify:

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

The answer is:

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