What is the logarithmic differentiation to evaluate for f(x)=(x)^ln(3) ?

Question

627 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-03-03T01:40:51

#f'(x)=ln(3)x^(ln(3)-1)#

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)#