Assignment: Simplify the derivative (with respect to x) of an integral (with respect to t), without calculating the integral.

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So I have been doing old exams for a calculus course and I encountered the problem in the attached image.
I have no clue how to approach it without calculating the integral first. I hope someone here does. Thanks in advance.
 

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So I have been doing old exams for a calculus course and I encountered the problem in the attached image.
I have no clue how to approach it without calculating the integral first. I hope someone here does. Thanks in advance.
Here is a template [imath]\displaystyle{D_x}\int_{{x^2}}^{{x^3}} {\sqrt {1 + \log (t)} dt = 3{x^2}\sqrt {1 + \log ({x^3})} - 2x\sqrt {1 + \log ({x^2})} } [/imath]
 
for [imath]u \text{ and } v[/imath] both functions of [imath]x[/imath]

[imath]\displaystyle \dfrac{d}{dx} \bigg[\int_u^v f(t) \, dt \bigg] = f(v) \cdot \dfrac{dv}{dt} - f(u) \cdot \dfrac{du}{dt}[/imath]
 
for [imath]u \text{ and } v[/imath] both functions of [imath]x[/imath]

[imath]\displaystyle \dfrac{d}{dx} \bigg[\int_u^v f(t) \, dt \bigg] = f(v) \cdot \dfrac{dv}{dt} - f(u) \cdot \dfrac{du}{dt}[/imath]
dv/dx, du/dx ?
 
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