How do you differentiate # x^sin(x)#?

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Answer 1 · 2016-10-19T08:05:54

#(dy)/(dx)=(x^sinx)(cosxlnx+sinx/x)#

let
#y=x^sinx#

take natural logarithms to both sides and simplify

#lny=lnx^sinx#

#=>lny=sinxlnx#

differentiate both sides wrt #x#

#d/(dx)(lny)=d/(dx)(sinxlnx)#

using implicit differentiation on the LHS; product rule on RHS

#=1/y(dy)/dx=cosxlnx+sinx/x#

#=>(dy)/(dx)=y(cosxlnx+sinx/x)#

substituting back for #y#

#(dy)/(dx)=(x^sinx)(cosxlnx+sinx/x)#