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Question
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
`(4x + 5sin x)/(3x + 7cos x)`
Solution
`d/dx(u/v) = (uv - uv)/v^2`
`d/dx((4x + 5sin x)/(3x + 7 cos x)) = ([d/dx(4x + 5sin x)](3x + 7 cos x) - (4" + 5 sin x) d/dx(3x + 7 cos x))/(3x + 7 cos x)^2`
= `((4 + 5 cos x)(3x + 7 cos x) - (4x + 5 sin x)(3 - 7 sin x))/(3x + 7 cos x)^2`
= `((12x + 28 cos x + 15x cos x + 35 cos^2 x) - (12x - 28x sin x + 15 sin x - 35 sin^2 x))/(3x + 7 cos x)^2`
= `(28(cos x + x sin x) + 15(x cos x - sin x) + 35 (cos^2 x + sin ^2 x))/(3x + 7 cos x)^2`
= `(35 + 15xcos x + 28 cos x + 28 x sin x - 15 sin x)/((3x + 7 cos x)^2)`
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