Derivative of e^(-cosx)

1. Let $u = - \cos{\left(x \right)}$.

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

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

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

      1. The derivative of cosine is negative sine:

         $$\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$$

      So, the result is: $\sin{\left(x \right)}$

   The result of the chain rule is:

   $$e^{- \cos{\left(x \right)}} \sin{\left(x \right)}$$

The answer is:

$$e^{- \cos{\left(x \right)}} \sin{\left(x \right)}$$