Find the derivative and simplify your
answer.

Ask by Mullins Floyd. in the United States

Mar 22,2025

Question

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

UpStudy AI · Accepted Answer

We begin with the integral
$$
I(x)=\int_{1}^{\ln x} \sqrt{t}\, dt.
$$
Since the upper limit is a function of $$ x $$, we use the Fundamental Theorem of Calculus together with the chain rule (Leibniz rule). If
$$
I(x)=\int_{a}^{g(x)} f(t)\, dt,
$$
then
$$
I'(x)=f(g(x))\cdot g'(x).
$$

In our case, we have:
- $$ f(t)=\sqrt{t} $$.
- $$ g(x)=\ln x $$ with $$ g'(x)=\frac{1}{x} $$.

Thus, substituting in, we get:
$$
\frac{d}{dx} \int_{1}^{\ln x} \sqrt{t}\, dt = \sqrt{\ln x}\cdot \frac{1}{x} = \frac{\sqrt{\ln x}}{x}.
$$
The derivative is $$ \frac{\sqrt{\ln x}}{x} $$.

Find the derivative and simplify your
answer.

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