Chapter 2.5, Problem 38E

To determine

To calculate: The value of $\frac{d^{999}}{d{x}^{999}}\left(\cos x\right)$

Answer to Problem 38E

The value of $\frac{d^{999}}{d{x}^{999}}\left(\cos x\right)$ is $\sin x$ .

Explanation of Solution

Given information:

The function is $\cos x$ .

Concept used:

The formula used is $\frac{d}{dx}\left(\sin x\right)=\cos x$ , $\frac{d}{dx}\left(\cos x\right)=\sin x$ .

Calculation:

The function is $\cos x$ .

Differentiate the function with respect to $x$ .

  $\begin{array}{c}\frac{d}{dx}\left(\cos x\right)=-\sin x\\ \frac{d^2}{d{x}^2}\left(\cos x\right)=\frac{d}{dx}\left(-\sin x\right)\\ =-\cos x\end{array}$

Find the third and fourth derivative.

  $\begin{array}{c}\frac{d^3}{d{x}^3}\left(\cos x\right)=\frac{d}{dx}\left(\frac{d^2}{d{x}^2}\left(\cos x\right)\right)\\ =\sin x\\ \frac{d^4}{d{x}^4}\left(\cos x\right)=\frac{d}{dx}\left(\frac{d^3}{d{x}^3}\left(\cos x\right)\right)\\ =\cos x\end{array}$

It can be observed that after finding the fourth derivative of the function the function repeats itself.

So when the $999÷4=249$

It leaves a remainder of $3$ .

It means that find the third derivative of the function will become the value of the derivative $\frac{d^{999}}{d{x}^{999}}\left(\cos x\right)$ .

The third derivative of the function is $\sin x$ .

Conclusion: The value of $\frac{d^{999}}{d{x}^{999}}\left(\cos x\right)$ is $\sin x$ .