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Question Number 59838 by necx1 last updated on 15/May/19

$$Whatisthenthderivativeofsinxin$$$$termsofthesinefunction?$$

Commented by maxmathsup by imad last updated on 15/May/19

$$iff\left(x\right)=sinx,f^{\left(n\right)}\left(x\right)=sin(x+\frac{n\pi }{2})withn⩾1and{f}^{\left(0\right)}=f$$$$letprovethisbyrecurrencewehave{f}^{\left(1\right)}\left(x\right)=cos\left(x\right)=sin(x+\frac{\pi }{2})$$$$letsupposetheequalitytrueattermn\Rightarrow f^{(n+1)}\left(x\right)={\left(f^{\left(n\right)}\left(x\right)\right)}^{{}^{\prime }}$$$$={\left(sin(x+\frac{n\pi }{2})\right)}^{{}^{\prime }}=cos(x+\frac{n\pi }{2})=sin(x+\frac{n\pi }{2}+\frac{\pi }{2})=sin(x+\frac{(n+1)\pi }{2})$$$$theresultisproved.$$

Answered by tanmay last updated on 15/May/19

$$y=sinx$$$$\frac{dy}{dx}=y_1=cosx=sin(\frac{\pi }{2}+x)$$$$y_2=-sinx=sin(2\times \frac{\pi }{2}+x)$$$$y_3=-cosx=sin(3\times \frac{\pi }{2}+x)$$$$y_4=sinx=sin(4\times \frac{\pi }{2}+x)$$$$y_n=sin(n\times \frac{\pi }{2}+x)$$$$heren=9$$$$y_9=sin(9\times \frac{\pi }{2}+x)$$

Commented by necx1 last updated on 15/May/19

$$oh.....Igetitnow.Thanks$$

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