Let #y=(ln(x))^sin(x)#
To find the derivative of such a problem we need to take logarithms on both the sides.
#ln(y)=ln((ln(x))^sin(x))#
This step is done to move the exponent to the front of the equation.
as #color(brown)(ln(a^n) = n*ln(a)#
#ln(y)=sin(x)*ln(ln(x))#
Now let us differentiate both sides with respect to #x#
We shall use product rule #color(blue)((uv)'=uv'+vu'#
#(1/y)dy/dx = sin(x)d/dx(ln(ln(x))) + ln(ln(x))d/dx(sin(x)#
#color(blue)"Derivative of ln(ln(x)) to be done using chain rule"#
#(1/y)dy/dx = sin(x)(1/ln(x))*d/dx(ln(x)+ln(ln(x))(cos(x))#
#(1/y)dy/dx =sin(x)(1/ln(x))*1/x+cos(x)ln(ln(x))#
#(1/y)dy/dx = sin(x)/(xln(x))+cos(x)ln(ln(x))#
#dy/dx = y{sin(x)/(xln(x))+cos(x)ln(ln(x))}#
#dy/dx = (ln(x))^sin(x){sin(x)/(xln(x))+cos(x)ln(ln(x))}#
