Question

Differentiate the following with respect to x.

(sin x)^{tan x}

Answer 1

Let y = (sin x)^{tan x}

Taking logarithm on both sides we get,

log y = tan x log(sin x)

Differentiating with respect to x,

`1/y "dy"/"dx" = tan x "d"/"dx"` (log (sin x)) + log (sin x) `"d"/"dx"` (tan x)

= tan x `1/(sin x)`(cos x) + log (sin x) sec²x

`1/y "dy"/"dx" = (sin x)/(cos x) xx (cos x)/(sin x)` + log (sin x)(sec²x)

= 1 + log (sin x)(sec²x)

`"dy"/"dx"` = y[1 + sec²x log (sin x)]

= (sin x)^{tan x}[1 + sec²x log (sin x)]