Differentiate $f(x)=\int_{\sqrt{x}}^{\arcsin x} \ln\theta d \theta$

What derivative rule sgould I use to differentiate
\[f(x)=\int_{\sqrt{x}}^{\arcsin x} \ln\theta d \theta.\] 


You need to use the Leibniz rule: 

Using this formula we get 

\[f'(x)=(\arcsin x)' \ln(\arcsin x)-(x^{1/2})'\ln(x^{1/2})\]
\[=\frac{1}{\sqrt{1-x^2}} \ln(\arcsin x)-\frac{1}{2}x^{-1/2}\ln (x^{1/2}).\]

Erdos Erdos
The answer is accepted.
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