Question

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

Answer 1

(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