Question

`"d"/"dx" [(cos x)^(log x)]` = ______.

पर्याय

• `(cos x)^(log x) (1/x log cos x + tan x)`

• `(cos x)^(log x) (1/x log cos x - tan x log x)`

• `(cos x)^(log x) (1/x log cos x + log x)`

• `(cos x)^(log x) (x log cos x - tan x log x)`

Answer 1

`"d"/"dx" [(cos x)^(log x)] = underline((cos x)^(log x) (1/x log cos x - tan x log x))`.

Explanation:

Let y = `(cos x)^(log x)`

Taking logarithm on both sides, we get

log y = log x log (cos x)

Differentiating both sides w.r.t.x, we get

`1/"y" * "dy"/"dx" = log x * 1/(cos x) * (- sin x) + log (cos x) * 1/x`

`= "dy"/"dx" = (cos x)^(log x) (1/x log cos x - tan x log x)`