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

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We must know the derivative of \cos x  .


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


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]

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