Derivative of sqrt(10x)+3sqrt(200-20x)

1. Differentiate $\sqrt{10 x} + 3 \sqrt{200 - 20 x}$ term by term:

   1. Let $u = 10 x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 10 x$:

      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: $10$

      The result of the chain rule is:

      $$\frac{\sqrt{10}}{2 \sqrt{x}}$$

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

      1. Let $u = 200 - 20 x$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(200 - 20 x\right)$:

         1. Differentiate $200 - 20 x$ term by term:

            1. The derivative of the constant $200$ is zero.

            2. 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: $-20$

            The result is: $-20$

         The result of the chain rule is:

         $$- \frac{10}{\sqrt{200 - 20 x}}$$

      So, the result is: $- \frac{30}{\sqrt{200 - 20 x}}$

   The result is: $- \frac{30}{\sqrt{200 - 20 x}} + \frac{\sqrt{10}}{2 \sqrt{x}}$

2. Now simplify:

   $$\frac{\sqrt{5} \left(- 3 \sqrt{x} + \frac{\sqrt{20 - 2 x}}{2}\right)}{\sqrt{x} \sqrt{10 - x}}$$

The answer is:

$$\frac{\sqrt{5} \left(- 3 \sqrt{x} + \frac{\sqrt{20 - 2 x}}{2}\right)}{\sqrt{x} \sqrt{10 - x}}$$