Question

y = (logx)^logx differentiate​

Answer 1

${ \rm{ \green \rightarrow { \underline{ \underline \orange{Answer}}} \green{ \leftarrow}}}$

$\small{ \bf{y = { log(x) }^{ log(x) } }} \\ \small{ \sf{Find \: the \: derivative \: of \: the \: function}} \\ \small{ \bf{ \to y' = \frac{d}{dx} ( { log(x) }^{ log(x) } ) }} \\ \small{ \sf{expand \: the \: expression}} \\ { \bf{ \to y' = \frac{d}{dx} ( {e}^{ ln( log(x)) log(x) } )}} \\ { \sf{Use \: differentiation \: rules}} \\ \small{ \bf{ \to y' = \frac{d}{dg}( {e}^{g}) \frac{d}{dx} ( ln( log(x) ) log(x) )}} \\ { \sf{Calculate \: the \: derivatives}} \\\small{\bf{ \to y' = {e}^{g} ( \frac{1}{ log(x)} \frac{1}{ ln( 10(x) ) } \times log(x) }} \\ \: \: \: \: \: \: \small{ \bf{ + ln( log(x) ) \times \frac{1}{ ln(10)x } )}} \\ { \sf{substitute \: back }} \\ \small{ \bf{ \to y'= ({e}^{ ln( log(x)) log(x)})( \frac{1}{ log(x) } \times \frac{1}{ ln(10)x } \times log(x) }} \\ \small{ \bf{ \: \: \: \: \: + ln( log(x \times \frac{1}{ ln(10)x } ) ) }} \\ \\ {\sf{Simplify\: the\: expression}} \\ \small{ \bf{ \to y' = \frac{ { log(x) }^{ log(x) } + { log(x) }^{ log(x) } + ln( log(x) ) }{ ln(10)x } }}$

Hence differentiated