Derivative of cos(2x)-e^(-5x)

1. Differentiate $\cos{\left(2 x \right)} - e^{- 5 x}$ term by term:

   1. Let $u = 2 x$.

   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 $\frac{d}{d x} 2 x$:

      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: $2$

      The result of the chain rule is:

      $$- 2 \sin{\left(2 x \right)}$$

   4. 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 $e^{u}$ is itself.

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

         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: $-5$

         The result of the chain rule is:

         $$- 5 e^{- 5 x}$$

      So, the result is: $5 e^{- 5 x}$

   The result is: $- 2 \sin{\left(2 x \right)} + 5 e^{- 5 x}$

The answer is:

$$- 2 \sin{\left(2 x \right)} + 5 e^{- 5 x}$$