Derivative of sqrt(2x-5)(x+4)

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = \sqrt{2 x - 5}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = 2 x - 5$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 5\right)$:

      1. Differentiate $2 x - 5$ 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: $2$

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

         The result is: $2$

      The result of the chain rule is:

      $$\frac{1}{\sqrt{2 x - 5}}$$

   $g{\left(x \right)} = x + 4$; to find $\frac{d}{d x} g{\left(x \right)}$:

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

      1. Apply the power rule: $x$ goes to $1$

      2. The derivative of the constant $4$ is zero.

      The result is: $1$

   The result is: $\frac{x + 4}{\sqrt{2 x - 5}} + \sqrt{2 x - 5}$

2. Now simplify:

   $$\frac{3 x - 1}{\sqrt{2 x - 5}}$$

The answer is:

$$\frac{3 x - 1}{\sqrt{2 x - 5}}$$