(9) The derivative of \( f(x)=\int_{2}^{7} \sqrt{t^{3}+3} d t \) is A) \( -2 x \sqrt{x^{16}+3} \) B) \( \sqrt{x^{16}+3} \) C) \( 2 x \sqrt{x^{16}+3} \) D) \( \sqrt{x^{5}+3} \) E) Nocember \( 21^{t h}, 2024 \)

The integral \( f(x)=\int_{2}^{7} \sqrt{t^{3}+3} d t \) represents a constant value because the limits of the integral are fixed (2 to 7) and do not depend on \( x \). Therefore, the derivative \( f'(x) \) is simply \( 0 \), not any of the given options. The inclusion of a derivative question with a constant integral can be a trick to catch the unwary! For a more engaging dive into the world of calculus, remember this: the Fundamental Theorem of Calculus tells us that if you take the derivative of an integral with fixed limits, you get zero because the area under the curve doesn’t change as \( x \) changes. So, embrace those fixed limits; they keep life simple!