Answer 1

Let y = `log[a^(cosx)/((x^2 - 3)^3 logx)]`

= loga^cosx – 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)`.