Questions: Use the formula (f'(x)=lim z rightarrow x fracf(z)-f(x)z-x) to find the derivative of the following function. (f(x)=5+sqrtx) (f'(x)= square ) (Type an exact answer, using radicals as needed.)

Solution Steps

Given the function: \[ f(x) = 5 + \sqrt{x} \]

Rewrite \(\sqrt{x}\) as \(x^{1/2}\): \[ f(x) = 5 + x^{1/2} \]

The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). Applying this to \(x^{1/2}\): \[ \frac{d}{dx} \left( x^{1/2} \right) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}} \]

Since the derivative of a constant is zero, the derivative of \(5\) is \(0\). Therefore, the derivative of the function \(f(x) = 5 + \sqrt{x}\) is: \[ f'(x) = 0 + \frac{1}{2\sqrt{x}} = \frac{1}{2\sqrt{x}} \]

Final Answer