Chapter 3.5, Problem 39E

To determine

To find: $(\frac{d^{725}}{d{x}^{725}})\sin x$

Answer to Problem 39E

  $(\frac{d^{725}}{d{x}^{725}})\sin x=co\mathrm{s}x$ .

Explanation of Solution

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

Calculation:Given equation is $\sin x$ .

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

Thus every fourth derivative returns us to our original function.

Observe that $725=181\times 4+1$ . This says that with $\frac{d^{725}}{d{x}^{725}}$ , the fourth derivative is taken $181$ times, which lands us back at $\sin x$ every time. There are then one more derivatives taken after landing at $\sin x$ for the $181$ time.

  $\frac{d}{dx}(\sin x)=\cos x$

Thus $(\frac{d^{725}}{d{x}^{725}})\sin x=co\mathrm{s}x$ .