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

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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 $x$ :

$Rightarrow frac(d)(dx)(ln(y))=frac(d)(dx)(ln(3) cdot ln(x))$

$Rightarrow frac(dy)(dx) cdot frac(d)(dy)(ln(y))=ln(3)frac(d)(dx)(ln(x))$

$Rightarrow frac(dy)(dx) cdot frac(1)(y)=frac(ln(3))(x)$

$Rightarrow frac(dy)(dx)=frac(yln(3))(x)$

$Rightarrow frac(dy)(dx)=frac(x^(ln(3)) cdot ln(3))(x)$

$Rightarrow frac(dy)(dx)=ln(3)x^(ln(3)-1)$

$therefore f'(x)=ln(3)x^(ln(3)-1)$