Derivative of exp(-x)+4/sqrt(x)

1. Differentiate $e^{- x} + \frac{4}{\sqrt{x}}$ term by term:

   1. Let $u = - x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- x\right)$:

      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: $-1$

      The result of the chain rule is:

      $$- e^{- x}$$

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

      1. Let $u = \sqrt{x}$.

      2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

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

         1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$

         The result of the chain rule is:

         $$- \frac{1}{2 x^{\frac{3}{2}}}$$

      So, the result is: $- \frac{2}{x^{\frac{3}{2}}}$

   The result is: $- e^{- x} - \frac{2}{x^{\frac{3}{2}}}$

The answer is:

$$- e^{- x} - \frac{2}{x^{\frac{3}{2}}}$$