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Question
Differentiate the following w.r.t.x:
`log[a^(cosx)/((x^2 - 3)^3 logx)]`
Solution
Let y = `log[a^(cosx)/((x^2 - 3)^3 logx)]`
= logacosx – log(x2 – 3)3 – log(log x)
= (cos x)(log a) – 3log(x2 – 3) –log(log x)
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[(cosx)(loga) - 3log(x^2 - 3) - log(logx)]`
`= (loga)."d"/"dx"(cosx) - 3"d"/"dx"[log(x^2 - 3)] - "d"/"dx"[log(logx)]`
`= (loga)(-sinx) - 3 xx (1)/(x^2 - 3)."d"/"dx"(x^2 - 3) - (1)/(logx)."d"/"dx"(logx)`
`= -(sinx)(loga) - (3)/(x^2 - 3) xx (2x - 0) - (1)/logx xx (1)/x`
`= -(sinx)(loga) - (6x)/(x^2 - 3) - (1)/(xlogx)`.
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