Put, $\cos \alpha = {3 \over 5},\sin \alpha = {4 \over 5}$ $\therefore {3 \over 5}\cos kx - {4 \over 5}\sin \,kx$ $= \cos \alpha .\cos kx - \sin \alpha .\sin kx$ $= \cos \left( {\alpha + kx} \right)$ So, $y = \sum\limits_{k = 1}^6 {k{{\cos }^{ - 1}}\left( {\cos \left( {\alpha + kx} \right)} \right)}$ $= \sum\limits_{k = 1}^6 {\left( {{k^2}x + kx} \right)}$ $\Rightarrow {{dy} \over {dx}} = \sum\limits_{k = 1}^6 {{k^2}}$ $= {{6 \times 7 \times 13} \over 6} = 91$