Daily math problems for July 8, 2026 — Derivatives, Trigonometric identities, Statistics

By the product rule, $f^{\mathrm{\prime }}(x)=(x^3{)}^{\mathrm{\prime }}\sin x+x^3(\sin x{)}^{\mathrm{\prime }}=3x^2\sin x+x^3\cos x$.

Using the closed forms $\cos {36}^{\circ }=\frac{1+\sqrt{5}}{4}$ and $\cos {72}^{\circ }=\frac{\sqrt{5}-1}{4}$, the difference is $\frac{(1+\sqrt{5})-(\sqrt{5}-1)}{4}=\frac{2}{4}=\frac{1}{2}$.

The slope is $8$ score points per additional hour. For a $0.5$-hour increase, the predicted change is $0.5\times 8=4$.