Find the derivative and simplify your answer. \[ \frac{d}{d x} \int_{1}^{\ln x} \sqrt{t} d t=\frac{\sqrt{[?]}}{} \]

To find the derivative, we can use the Fundamental Theorem of Calculus, which tells us that: \[ \frac{d}{dx} \int_{a}^{g(x)} f(t) dt = f(g(x)) \cdot g'(x) \] In your case, \( f(t) = \sqrt{t} \) and \( g(x) = \ln x \). First, we need to compute \( f(g(x)) \): \[ f(g(x)) = f(\ln x) = \sqrt{\ln x} \] Next, we differentiate \( g(x) = \ln x \): \[ g'(x) = \frac{1}{x} \] Now we can apply the theorem: \[ \frac{d}{dx} \int_{1}^{\ln x} \sqrt{t} dt = \sqrt{\ln x} \cdot \frac{1}{x} \] Thus, the derivative simplifies to: \[ \frac{\sqrt{\ln x}}{x} \] So the final answer you are looking for is: \[ \frac{d}{d x} \int_{1}^{\ln x} \sqrt{t} d t = \frac{\sqrt{\ln x}}{x} \] And, filling in the placeholders: \[ \frac{\sqrt{[\ln x]}}{x} \]