Derivative of sqrt(7,7x+5)

1. Let $u = \frac{77 x}{10} + 5$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\frac{77 x}{10} + 5\right)$:

   1. Differentiate $\frac{77 x}{10} + 5$ 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: $\frac{77}{10}$

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

      The result is: $\frac{77}{10}$

   The result of the chain rule is:

   $$\frac{77}{20 \sqrt{\frac{77 x}{10} + 5}}$$

4. Now simplify:

   $$\frac{77 \sqrt{10}}{20 \sqrt{77 x + 50}}$$

The answer is:

$$\frac{77 \sqrt{10}}{20 \sqrt{77 x + 50}}$$