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Question
Differentiate the following w.r.t.x:
`log(sqrt((1 + cos((5x)/2))/(1 - cos((5x)/2))))`
Solution
Using `log(a/b)` = log a − log b
log ab = b log a
`y = log(sqrt(1 + cos ((5x)/2))) - log(sqrt(1 - cos ((5x)/2)))`
`y = log[1 + cos ((5x)/2)]^(1/2) - log[1 - cos((5x)/2)]^(1/2)`
`y = (1)/(2)log[1 + cos((5x)/2)] - (1)/(2)log[(1 - cos((5x)/2)]`
Differentiating w.r.t. x
`"dy"/"dx" = 1/2 × 1/(1 + cos((5x)/2)) × "d"/"dx"(1 + cos (5x)/2) - 1/2 × 1/(1 - cos((5x)/2)) × "d"/"dx"(1 - cos (5x)/(2))`
`"dy"/"dx" = 1/2 × 1/(1 + cos((5x)/2)) × [0 - sin ((5x)/2)] . 5/2 "d"/"dx" x - 1/2 × 1/(1 - cos((5x)/2)) × [0 + sin ((5x)/2)] . 5/2 "d"/"dx" x`
`"dy"/"dx" = 1/2 × 1/(1 + cos((5x)/2)) × - sin ((5x)/2) . 5/2 - 1/2 × 1/(1 - cos((5x)/2)) × sin ((5x)/2) . 5/2`
`"dy"/"dx" = [- 5sin((5x)/2)]/[4(1 + cos((5x)/2))] - [5sin((5x)/2)]/[4(1 - cos((5x)/2))]`
`"dy"/"dx" = [- 5sin((5x)/2)]/4. [1/(1 + cos((5x)/(2))) + 1/(1 - cos((5x)/(2)))]`
`"dy"/"dx" = [- 5sin((5x)/2)]/4. [(1 - cos ((5x)/2) + 1 + cos ((5x)/2)]/(1^2 - cos^2 ((5x)/2))]`
`"dy"/"dx" = [- 5sin((5x)/2)]/4. 2/(sin^2((5x)/2))` ...[ ∵ 1 – cos2x = sin2x]
`"dy"/"dx" = - 5/2 . 1/(sin((5x)/2))`
`"dy"/"dx" = - 5/2 . "cosec" ((5x)/2)`
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