Derivative of 3e^(3x)+3cos(3x)

1. Differentiate $3 e^{3 x} + 3 \cos{\left(3 x \right)}$ term by term:

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

      1. Let $u = 3 x$.

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

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

         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: $3$

         The result of the chain rule is:

         $$3 e^{3 x}$$

      So, the result is: $9 e^{3 x}$

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

      1. Let $u = 3 x$.

      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 $\frac{d}{d x} 3 x$:

         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: $3$

         The result of the chain rule is:

         $$- 3 \sin{\left(3 x \right)}$$

      So, the result is: $- 9 \sin{\left(3 x \right)}$

   The result is: $9 e^{3 x} - 9 \sin{\left(3 x \right)}$

The answer is:

$$9 e^{3 x} - 9 \sin{\left(3 x \right)}$$