Derivative of y=sqrt(2ax-x^2)

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

   1. Let $u = 2 a x - x^{2}$.

   2. Apply the power rule: $u^{2}$ goes to $2 u$

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

      1. Differentiate $2 a x - x^{2}$ 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: $2 a$

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

            1. Apply the power rule: $x^{2}$ goes to $2 x$

            So, the result is: $- 2 x$

         The result is: $2 a - 2 x$

      The result of the chain rule is:

      $$\left(2 a - 2 x\right) \left(4 a x - 2 x^{2}\right)$$

   So, the result is: $t \left(2 a - 2 x\right) \left(4 a x - 2 x^{2}\right)$

2. Now simplify:

   $$4 t x \left(a - x\right) \left(2 a - x\right)$$

The answer is:

$$4 t x \left(a - x\right) \left(2 a - x\right)$$