Derivative of 4/sqrt(4x+1)

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

   1. Let $u = \sqrt{4 x + 1}$.

   2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{4 x + 1}$:

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

      2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$

      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:

         $$\frac{2}{\sqrt{4 x + 1}}$$

      The result of the chain rule is:

      $$- \frac{2}{\left(4 x + 1\right)^{\frac{3}{2}}}$$

   So, the result is: $- \frac{8}{\left(4 x + 1\right)^{\frac{3}{2}}}$

2. Now simplify:

   $$- \frac{8}{\left(4 x + 1\right)^{\frac{3}{2}}}$$

The answer is:

$$- \frac{8}{\left(4 x + 1\right)^{\frac{3}{2}}}$$