Derivative of 1/(4x+1)

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

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

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

   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:

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

4. Now simplify:

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

The answer is:

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