Question

The first derivative of the function \[ U(r)=4\left[\left(\frac{1}{r}\right)^{12}-\left(\frac{1}{r}\right)^{6}\right] \] evaluated at \(r=1\) is ______________ (in integer).

Hint:

Always convert reciprocal powers into \(r^{-n}\) before differentiation.
For \(U(r)=Ar^{-n}\), derivative is \(-nAr^{-n-1}\).

Answer 1

Correct Answer: -24

Step 1: Simplify function
\[ U(r)=4\left(r^{-12}-r^{-6}\right). \] Step 2: Differentiate w.r.t \(r\)
\[ U'(r)=4\left(-12r^{-13}+6r^{-7}\right) =-48r^{-13}+24r^{-7}. \] Step 3: Evaluate at \(r=1\)
\[ U'(1)=-48(1)^{-13}+24(1)^{-7}=-48+24=-24. \] Thus the derivative at \(r=1\) is \(\boxed{-24}\).