Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

Question Number 108350 by Ar Brandon last updated on 16/Aug/20

Calculate (d^n /dx^n )(sinx) , (d^n /dx^n )(lnx)

$$\mathrm{Calculate}\:\frac{\mathrm{d}^{\mathrm{n}} }{\mathrm{dx}^{\mathrm{n}} }\left(\mathrm{sinx}\right)\:,\:\frac{\mathrm{d}^{\mathrm{n}} }{\mathrm{dx}^{\mathrm{n}} }\left(\mathrm{lnx}\right) \\ $$

Answered by Dwaipayan Shikari last updated on 16/Aug/20

f(x)=logx f′(x)=(1/x) f′′(x)=−(1/x^2 ) f′′′(x)=(2/x^3 ) f^((iv)) =−(6/x^4 ) f^((v)) =((24)/x^5 ) ...f^((n)) =(−1)^(n+1) .(((n−1)!)/x^n )

$${f}\left({x}\right)={logx} \\ $$$${f}'\left({x}\right)=\frac{\mathrm{1}}{{x}} \\ $$$${f}''\left({x}\right)=−\frac{\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$$${f}'''\left({x}\right)=\frac{\mathrm{2}}{{x}^{\mathrm{3}} } \\ $$$${f}^{\left({iv}\right)} =−\frac{\mathrm{6}}{{x}^{\mathrm{4}} } \\ $$$${f}^{\left({v}\right)} =\frac{\mathrm{24}}{{x}^{\mathrm{5}} } \\ $$$$...{f}^{\left({n}\right)} =\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} .\frac{\left({n}−\mathrm{1}\right)!}{{x}^{{n}} } \\ $$

Answered by ajfour last updated on 16/Aug/20

y=sin x (dy/dx)=y_1 =sin ((π/2)+x) y_2 =sin [2((π/2))+x] .... y_n =sin [n((π/2))+x]

$${y}=\mathrm{sin}\:{x} \\ $$$$\frac{{dy}}{{dx}}={y}_{\mathrm{1}} =\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}+{x}\right) \\ $$$${y}_{\mathrm{2}} =\mathrm{sin}\:\left[\mathrm{2}\left(\frac{\pi}{\mathrm{2}}\right)+{x}\right] \\ $$$$.... \\ $$$${y}_{{n}} =\mathrm{sin}\:\left[{n}\left(\frac{\pi}{\mathrm{2}}\right)+{x}\right] \\ $$

Answered by mathmax by abdo last updated on 17/Aug/20

(d^n /dx^n )(sinx) =sin(x+((nπ)/2)) we can prove this by recurrence (d/dx)(lnx) =(1/x) ⇒(d^n /dx^n )(lnx) =((1/x))^((n−1)) =(((−1)^(n−1) (n−1)!)/x^n ) for n≥1

$$\frac{\mathrm{d}^{\mathrm{n}} }{\mathrm{dx}^{\mathrm{n}} }\left(\mathrm{sinx}\right)\:=\mathrm{sin}\left(\mathrm{x}+\frac{\mathrm{n}\pi}{\mathrm{2}}\right)\:\mathrm{we}\:\mathrm{can}\:\mathrm{prove}\:\mathrm{this}\:\mathrm{by}\:\mathrm{recurrence} \\ $$$$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{lnx}\right)\:=\frac{\mathrm{1}}{\mathrm{x}}\:\Rightarrow\frac{\mathrm{d}^{\mathrm{n}} }{\mathrm{dx}^{\mathrm{n}} }\left(\mathrm{lnx}\right)\:=\left(\frac{\mathrm{1}}{\mathrm{x}}\right)^{\left(\mathrm{n}−\mathrm{1}\right)} \:=\frac{\left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} \left(\mathrm{n}−\mathrm{1}\right)!}{\mathrm{x}^{\mathrm{n}} } \\ $$$$\mathrm{for}\:\mathrm{n}\geqslant\mathrm{1} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com