Question If $F(x)=\int_{2}^{4 x^{5}} \frac{2}{\ln \left(t^{2}\right)} d t$, what is $F^{\prime}(x)$ ? (Do not include " $F^{\prime}(x)=$ " in your answer.) Provide your answer below:

Step 1 ：The problem is asking for the derivative of the function \(F(x) = \int_{2}^{4x^5} \frac{2}{\ln(t^2)} dt\).

Step 2 ：This is a problem of the Fundamental Theorem of Calculus part 1, which states that if a function \(f\) is continuous over the interval \([a, b]\) and \(F\) is an antiderivative of \(f\) on \([a, b]\), then \(\int_{a}^{b} f(x) dx = F(b) - F(a)\).

Step 3 ：To find \(F'(x)\), we need to differentiate the integral with respect to \(x\).

Step 4 ：The derivative of the integral of a function is just the function itself, evaluated at the upper limit of the integral, times the derivative of the upper limit with respect to \(x\).

Step 5 ：So, \(F'(x) = \frac{2}{\ln((4x^5)^2)} * \frac{d}{dx}(4x^5)\).

Step 6 ：Let's calculate this.

Step 7 ：The derivative of the function \(F(x)\) is \(\boxed{\frac{40x^{4}}{\ln(16x^{10})}}\).