Derivative of 2*e^x/x

1. Apply the quotient rule, which is:

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

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

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

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

      1. The derivative of $e^{x}$ is itself.

      So, the result is: $2 e^{x}$

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

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

   Now plug in to the quotient rule:

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

2. Now simplify:

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

The answer is:

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