Chapter 3.5, Problem 38E

To determine

To find: $(\frac{d^{999}}{d{x}^{999}})\cos x$

Answer to Problem 38E

  $(\frac{d^{999}}{d{x}^{999}})\cos x=\sin x$ .

Explanation of Solution

Given information: Given equation is $\cos x$ .

Calculation:Given equation is $\cos x$ .

  $\begin{array}{l}\frac{d}{dx}\cos x=-\sin x=(\frac{d^1}{d{x}^1})\cos x\\ \frac{d}{dx}(-\sin x)=-\cos x=(\frac{d^2}{d{x}^2})\cos x\\ \frac{d}{dx}(-\cos x)=\sin x=(\frac{d^3}{d{x}^3})\cos x\\ \frac{d}{dx}(\sin x)=\cos x=(\frac{d^4}{d{x}^4})\cos x\end{array}$

Thus every fourth derivative returns us to our original function.

Observe that $999=249\times 4+3$ . This says that with $\frac{d^{999}}{d{x}^{999}}$ the fourth derivative is taken $249$ times, which lands us back at $\cos x$ every time. There are then three more derivatives taken after landing at $\cos x$ for the $249$ time.

  $(\frac{d^3}{d{x}^3})\cos x=\sin x$

Thus $(\frac{d^{999}}{d{x}^{999}})\cos x=\sin x$ .