Derivative of x*lnx-e^-0,5x^2

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

   1. Apply the product rule:

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

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

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

      $g{\left(x \right)} = \log{\left(x \right)}$; 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: $\log{\left(x \right)} + 1$

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

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

      So, the result is: $- \frac{2 x}{e^{\frac{1}{2}}}$

   The result is: $- \frac{2 x}{e^{\frac{1}{2}}} + \log{\left(x \right)} + 1$

The answer is:

$$- \frac{2 x}{e^{\frac{1}{2}}} + \log{\left(x \right)} + 1$$