y=(tanx)^(1/x)

ddx((tan(x))1x)

=(tan(x))1x⋅ddx(ln(tan(x))⋅1x)

=ddx(ln(tan(x))x)⋅(tan(x))1x

=x⋅ddx(ln(tan(x)))−ddx(x)⋅ln(tan(x))x2⋅(tan(x))1x

=(tan(x))1x⋅(1tan(x)⋅ddx(tan(x))⋅x−1⋅ln(tan(x)))x2

=(tan(x))1x⋅((sec(x))2⋅xtan(x)−ln(tan(x)))x2

=(tan(x))1x⋅(x⋅(sec(x))2tan(x)−ln(tan(x)))x2

Rewrite/simplify:

=(tan(x))1x⋅((sec(x))2x⋅tan(x)−ln(tan(x))x2)

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