Find `(dy)/(dx)` when
(i) `Y = (x)/(log_(a)X)`
(ii) Y = cos x cos 2x cos 4x cos 8x
(iii) `y = x^(3) + tan x - e^(x) + x log_(e) x + ("sin" x) a^(x) + cossec x - 2007, a gt 0`
(iv) `y = (1 + x) (1 + x^(2)) (1 + x^(4)) (1 + x^(8))`……`(1 + x^(2n))`
(v) If y = f(x), where `f (x + (1)/(x)) = x^(4) + x^(-4)`

Explanation

We have
(i) `y = (x)/(log_(a) X) = (x)/(log_(e) x xx log_(a) e) = (1)/(log_(a) e) ((x)/(log_(e) x))`
`implies (dy)/(dx) = (1)/(log_(a) e) (d)/(dx) ((x)/(log_(e)x)) = (1)/(log_(a)e) [((dx)/(dx) xx log_(e) x - x (d)/(dx) (log_(e) x))/((log_(e) x)^(2))]`
`= ((log_(e) a) [log_(e) x - x (1)/(x)])/((log_(e) x)^(2)) = ((log_(e) a) log_(e) ((x)/(e)))/((log_(e) x)^(2))`
(ii) Y = cos x cos 2x cos 4x cos 8 x
`= (2 sin x)/(2 sin x) cos x cox 2x cos 4x cos 8 x`
`= (2 sin 2x cos 2x cos 4x cos 8x)/(2 xx 2 sin x)`
`= (2 sin 4 xx cos 4 xx cos 8 x)/(2 xx 4 sin x)`
`= (2 sin 4x cos 4x cos 8x)/(2 xx 4 sin x)`
`= (2 sin 8 x cos 8 x)/(2 xx 8 sin x)`
`= (sin 16 x)/(16 sin x)`
`implies (dy)/(dx) = (d)/(dx) ((sin 16 x)/(16 sin x)) = (1)/(16) (sin xx (d)/(dx)(sin 16x) - sin 16x (d)/(dx) (sinx))/(sin^(2) x)`
`= (1)/(16) (sin x 16 cos 16x - sin 16 x cos x)/(sin^(2) x)`
`=(1)/(16) [(16 sin x cos 16 x - sin 16 x cos x)/(sin^(2) x)]`
(iii) `y = x^(3) + tan x - e^(x) + x log_(e) x + (sin x) a^(x) + cosecx - 2007`
`implies (dy)/(dx) = (dx^(3))/(dx) + (d)/(dx) (tan x) - (d)/(dx) (e^(x)) + (d)/(dx) (x log_(e) x) + (d)/(dx) (a^(x) sin x) + (d)/(dx) (cosec x) - (d)/(dx) (2007)`
`implies (dy)/(dx) = 3x^(2) + sec^(2) x - e^(x) + (dx)/(dx) xx log_(e) x + x (d)/(dx) (log_(e)^(x)) + (da^(x))/(dx) xx sin x + a^(x) (d sin x)/(dx) + (- cosec x cot x) - 0`
`= 3x^(2) + sec^(2) x - e^(x) + log_(3) x + x (1)/(x) + a^(x) sin x log_(e) a + a^(x) cos x - cosecx cot x`
`= 3x^(2) + sec^(2) x - cosec x cot x - x - e^(x) log_(x) x + a^(x) sin x log_(e) a +a^(x) cos x + 1`
(iv) `Y = (1 + x)(1 + x^(2)) (1 + x^(4))(1 + x^(8)) .....(1 + x^(2n))`
`= ((1 - x)/(1 - x)) (1 + x)(1 + x^(2)) (1 + x^(4))(1 + x^(8)).....(1 + 2^(2n))`
`= ((1 - x^(2)) (1 + x^(2))(1 + x^(4))(1 + x^(8))...(1 + x^(2n)))/((1 - x))`
`= ((1- x^(4))(1 + x^(4))(1 + x^(8))...(1 + x^(2n)))/((1-x))`
`= ((1 - x^(8))(1 + x^(8))....(1 + x^(2n)))/((1 - x))`
`y = ((1 - x^(2n))(1 + x^(8))...(1 + x^(2^(n + 1))))/(1 -x)`
`implies ((1 - x) (d)/(dx) (1 - x^(2^(n+1))) - (1 - x^(n^(n +1))) (d)/(dx) (1 - x))/((1 - x)^(2))`
`= ((1 -x)(0 - 2^(n + 1).X^(n+1))-(1 - x^(2^(n + 1)))(0-1))/((1 - x)^(2))`
`= ((1 - x^(2^(n + 1))) - 2^(n _ 1) (1 - x) (x^(2^(n + 1))))/((1 - x)^(2))`
We have
`f(x + (1)/(x)) = x^(4) + (1)/(x^(4)) = (x^(2) + (1)/(x^(2)))^(2) - 2`
`= [(x + (1)/(x))^(2) - 2]^(2) - 2`
`implies f(x) = (x^(2) -2)^(2) - 2 = x^(4) - 4x^(2) + 2 = y`
`implies (dy)/dx) = (d)/(dx) (x^(4)) - (d)/(dx) (4x^(2)) + (d)/(dx) (2)`
`= 4x^(3) - 8x + 0`
`= 4x (x^(2) - 2)`