QuestionFind the derivative of the function. $$\begin{array}{l} f(x)=\ln (\sqrt{x}+5) \\ f^{\prime}(x)=\square \end{array}$$

Studdy Solution
Apply the chain rule to find $f'(x)$:
The chain rule states:
$$\frac{d}{dx}[f(x)] = \frac{d}{du}[\ln(u)] \cdot \frac{d}{dx}[u]$$
Substitute the derivatives:
$$f'(x) = \frac{1}{u} \cdot \frac{1}{2\sqrt{x}}$$
Substitute back $u = \sqrt{x} + 5$:
$$f'(x) = \frac{1}{\sqrt{x} + 5} \cdot \frac{1}{2\sqrt{x}}$$
Simplify the expression:
$$f'(x) = \frac{1}{2\sqrt{x}(\sqrt{x} + 5)}$$
The derivative of the function is:
$$f'(x) = \frac{1}{2\sqrt{x}(\sqrt{x} + 5)}$$