Find the derivative of the function using the definition of derivative. \[ g(x)=\sqrt{4-x} \] \[ g^{\prime}(x)= \] State the domain of the function. (Enter your answer using interval notation.) State the domain of its derivative. (Enter your answer using interval notation.)

Step 1 ：First, we find the derivative of the function using the definition of derivative.

Step 2 ：\(g^{\prime}(x) = \lim_{h\to 0} \frac{g(x+h) - g(x)}{h}\)

Step 3 ：Substitute \(g(x)\) and \(g(x+h)\) into the equation.

Step 4 ：\(g^{\prime}(x) = \lim_{h\to 0} \frac{\sqrt{4-(x+h)} - \sqrt{4-x}}{h}\)

Step 5 ：Use the conjugate to simplify the numerator.

Step 6 ：\(g^{\prime}(x) = \lim_{h\to 0} \frac{h}{h(\sqrt{4-(x+h)} + \sqrt{4-x})}\)

Step 7 ：Cancel out the \(h\) in the numerator and denominator.

Step 8 ：\(g^{\prime}(x) = \lim_{h\to 0} \frac{1}{\sqrt{4-(x+h)} + \sqrt{4-x}}\)

Step 9 ：As \(h\) approaches 0, the derivative becomes.

Step 10 ：\(g^{\prime}(x) = \frac{1}{2\sqrt{4-x}}\)

Step 11 ：Next, we find the domain of the function and its derivative.

Step 12 ：For the function \(g(x)\), the expression under the square root must be greater than or equal to 0.

Step 13 ：\(4 - x \ge 0\)

Step 14 ：\(x \le 4\)

Step 15 ：\(\text{Domain of } g(x): (-\infty, 4]\)

Step 16 ：For the derivative \(g^{\prime}(x)\), the expression under the square root and in the denominator must be greater than 0.

Step 17 ：\(4 - x > 0\)

Step 18 ：\(x < 4\)

Step 19 ：\(\text{Domain of } g^{\prime}(x): (-\infty, 4)\)