Derivative of y=e^-cos5x

1. Let $u = - \cos{\left(5 x \right)}$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- \cos{\left(5 x \right)}\right)$:

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

      1. Let $u = 5 x$.

      2. The derivative of cosine is negative sine:

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$:

         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: $5$

         The result of the chain rule is:

         $$- 5 \sin{\left(5 x \right)}$$

      So, the result is: $5 \sin{\left(5 x \right)}$

   The result of the chain rule is:

   $$5 e^{- \cos{\left(5 x \right)}} \sin{\left(5 x \right)}$$

The answer is: