Derivative of sin(4x+1)

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

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 $\frac{d}{d x} \left(4 x + 1\right)$:

   1. Differentiate $4 x + 1$ 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 $1$ is zero.

      The result is: $4$

   The result of the chain rule is:

   $$4 \cos{\left(4 x + 1 \right)}$$

4. Now simplify:

   $$4 \cos{\left(4 x + 1 \right)}$$

The answer is:

$$4 \cos{\left(4 x + 1 \right)}$$