Question Number 82059 by jagoll last updated on 18/Feb/20
$${what}\:{is}\:{derivative}\:{of}\:\:{h}\:=\:\sqrt{{ln}\left({x}\right)} \\ $$$${by}\:{first}\:{principle}\:{method}\: \\ $$
Answered by Henri Boucatchou last updated on 18/Feb/20
$$\frac{\mathrm{dh}\left(\mathrm{x}\right)}{\mathrm{dx}}=\frac{\mathrm{d}\sqrt{\mathrm{lnx}}}{\mathrm{dx}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\frac{\mathrm{dlnx}}{\mathrm{dx}}}{\mathrm{2}\sqrt{\mathrm{lnx}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{2x}\sqrt{\mathrm{lnx}}} \\ $$
Answered by mr W last updated on 18/Feb/20
![h′=lim_(Δx→0) (((√(ln (x+Δx)))−(√(ln x)))/(Δx)) =lim_(Δx→0) (((√(ln x+ln (1+((Δx)/x))))−(√(ln x)))/(Δx)) =(√(ln x))lim_(Δx→0) (((√(1+((ln (1+((Δx)/x)))/(ln x))))−1)/(Δx)) =(√(ln x))lim_(Δx→0) (({1+(1/2)[((ln (1+((Δx)/x)))/(ln x))]+...}−1)/(Δx)) =(√(ln x))lim_(Δx→0) {(1/2)[((ln (1+((Δx)/x)))/(Δx ln x))]+...} =(√(ln x))lim_(Δx→0) {(1/2)[((ln (1+((Δx)/x))^(x/(Δx)) )/(xln x))]+...} =(√(ln x))×(1/2)×((ln e)/(xln x)) =(1/(2x(√(ln x))))](https://www.tinkutara.com/question/Q82108.png)
$${h}'=\underset{\Delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{\mathrm{ln}\:\left({x}+\Delta{x}\right)}−\sqrt{\mathrm{ln}\:{x}}}{\Delta{x}} \\ $$$$=\underset{\Delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{\mathrm{ln}\:{x}+\mathrm{ln}\:\left(\mathrm{1}+\frac{\Delta{x}}{{x}}\right)}−\sqrt{\mathrm{ln}\:{x}}}{\Delta{x}} \\ $$$$=\sqrt{\mathrm{ln}\:{x}}\underset{\Delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{\mathrm{1}+\frac{\mathrm{ln}\:\left(\mathrm{1}+\frac{\Delta{x}}{{x}}\right)}{\mathrm{ln}\:{x}}}−\mathrm{1}}{\Delta{x}} \\ $$$$=\sqrt{\mathrm{ln}\:{x}}\underset{\Delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left\{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{\mathrm{ln}\:\left(\mathrm{1}+\frac{\Delta{x}}{{x}}\right)}{\mathrm{ln}\:{x}}\right]+…\right\}−\mathrm{1}}{\Delta{x}} \\ $$$$=\sqrt{\mathrm{ln}\:{x}}\underset{\Delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left\{\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{\mathrm{ln}\:\left(\mathrm{1}+\frac{\Delta{x}}{{x}}\right)}{\Delta{x}\:\mathrm{ln}\:{x}}\right]+…\right\} \\ $$$$=\sqrt{\mathrm{ln}\:{x}}\underset{\Delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left\{\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{\mathrm{ln}\:\left(\mathrm{1}+\frac{\Delta{x}}{{x}}\right)^{\frac{{x}}{\Delta{x}}} }{{x}\mathrm{ln}\:{x}}\right]+…\right\} \\ $$$$=\sqrt{\mathrm{ln}\:{x}}×\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{ln}\:{e}}{{x}\mathrm{ln}\:{x}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{x}\sqrt{\mathrm{ln}\:{x}}} \\ $$