Derivative of sin(4x+5)+e^(3x)

1. Differentiate $e^{3 x} + \sin{\left(4 x + 5 \right)}$ term by term:

   1. Let $u = 4 x + 5$.

   2. The derivative of sine is cosine:

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x + 5\right)$:

      1. Differentiate $4 x + 5$ term by term:

         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: $4$

         2. The derivative of the constant $5$ is zero.

         The result is: $4$

      The result of the chain rule is:

      $$4 \cos{\left(4 x + 5 \right)}$$

   4. Let $u = 3 x$.

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

   6. 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:

      $$3 e^{3 x}$$

   The result is: $3 e^{3 x} + 4 \cos{\left(4 x + 5 \right)}$

2. Now simplify:

   $$3 e^{3 x} + 4 \cos{\left(4 x + 5 \right)}$$

The answer is:

$$3 e^{3 x} + 4 \cos{\left(4 x + 5 \right)}$$