We have: #f(x)=x^(ln(3)) Rightarrow y=x^(ln(3))#
Let's apply #ln# to both sides of the equation:
#Rightarrow ln(y)=ln(x^(ln(3)))#
#Rightarrow ln(y)=ln(3) cdot ln(x)#
Then, let's differentiate both sides with respect to #x#:
#Rightarrow frac(d)(dx)(ln(y))=frac(d)(dx)(ln(3) cdot ln(x))#
#Rightarrow frac(dy)(dx) cdot frac(d)(dy)(ln(y))=ln(3)frac(d)(dx)(ln(x))#
#Rightarrow frac(dy)(dx) cdot frac(1)(y)=frac(ln(3))(x)#
#Rightarrow frac(dy)(dx)=frac(yln(3))(x)#
#Rightarrow frac(dy)(dx)=frac(x^(ln(3)) cdot ln(3))(x)#
#Rightarrow frac(dy)(dx)=ln(3)x^(ln(3)-1)#
#therefore f'(x)=ln(3)x^(ln(3)-1)#
