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 ：$g^{\prime }(x)=\underset{h\to 0}{lim}\frac{g(x+h)-g(x)}{h}$

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

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

Step 5 ：Use the conjugate to simplify the numerator.

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

Step 7 ：Cancel out the $h$ in the numerator and denominator.

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

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

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