Answer to Question #188397 in Economics of Enterprise for faryal

Question #188397

Differentiate the following logarithmic functions and show that 𝑦=𝑙𝑛(𝑥+5)2≠[ln⁡(𝑥+5)]2


1
Expert's answer
2021-05-04T11:17:26-0400

(i)

"y=ln(x+5)^2"


"\\frac{dy}{dn}=\\frac{d}{dn}ln(x+5)^2"


"Rule[lna^b=blna]"


"\\frac{dy}{dn}=\\frac{d}{dn}2ln(x+5)"


"=2\\frac{d}{dn}ln(x+5)"


"Rule:[\\frac{d}{dn}inx=\\frac{1}{n}]"


"\\frac{dy}{dn}=2(\\frac{1}{x+5}).\\frac{d}{dn}(x+5)"


"=\\frac{2}{n+5}"


"\\frac{dy}{dn}=\\frac{2}{n+5}...........................................(i)"


(ii)

"y=[ln(x+5)]^2"


"\\frac{dy}{dn}=\\frac{d}{dn}[ln(n+5)]^2"


"=2ln(n+5).\\frac{d}{dn}ln(x+5)"


"=2ln(x+5).\\frac{1}{n+5}(x+5)"


"\\frac{dy}{dn}=\\frac{2}{x+5}ln(x+5).....................................(ii)"


The derivatives of the two functions are different so the two functions are not same


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