Problem

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.)

Solution

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

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