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

Question

82109 views around the world

You can reuse this answer
Creative Commons License

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 $y$

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

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