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#### Explain solution RD Sharma class 12 Chapter 11 Higher Order Derivatives Exercise Multiple Choice Question question 4

(b)

Hint:

We must know the derivative of $\cos x$  .

Given:

$= \frac{d^{20}}{dx^{20}}\left ( 2 \cos x \cos 3x \right )$

Explanation:

First let us solve the inner function,

$= \frac{d}{dx}\left ( 2 \cos x \cos 3x \right )$

Using identity  $\left ( 2 \cos x \cos 3x \right )$

\begin{aligned} &=\cos (-2 x)+\cos (4 x) \\ & \end{aligned}

$=\cos (2 x)+\cos (4 x)$

So, now we have to compute $20^{th}$ derivative of  $\cos \left ( 2x \right )+\cos\left ( 4x \right )$

Since, successive derivative of $\cos x$  cycle in  $4:-\sin x,-\cos x, \sin x, \cos x\cdot \cdot \cdot$

$4^{th}$ derivative of $\cos x$  is $20=\left ( 5.4 \right )^{th}$  derivative of  $\cos x$  is also $\cos x$ .

Keep chain in mind,

\begin{aligned} & & 2^{20} \cos (2 x)+4^{20} \cos (4 x) \\ \end{aligned}

$\Rightarrow \quad 2^{20}\left[\cos (2 x)+2^{20} \cos (4 x)\right]$