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Derivati i funksionit te rendit te dyte

derivati i rendit te dyte

Derivati i funksionit te rendit te dyte

Në qoftë se ekziston derivati i funksionit f’ ku \displaystyle f':y=f'\left( x \right) në pikën \displaystyle x\in I, ai quhet derivat i dytë (ose derivat i rendit II) i funksionit f në pikën x dhe shënohet \displaystyle f''\left( x \right)

 

 

 

Derivatet e funksioneve y=tgx,  y=cotgx

Teoremë: Në çdo pikë \displaystyle x\in R ku \displaystyle \cos x\ne 0, funksioni y=tgx ka derivate dhe ky është \displaystyle \frac{1}{{{\cos }^{2}}x}.

Pra, \displaystyle \left( tgx \right)'=\frac{1}{{{\cos }^{2}}x}.

 

Rrjedhim: Në çdo pikë \displaystyle x\in R ku \displaystyle \sin x\ne 0, funksioni y=cotgx ka derivate dhe ai është \displaystyle -\frac{1}{{{\sin }^{2}}x}.

Pra, \displaystyle \left( cotgx \right)'=-\frac{1}{{{\sin }^{2}}x}.

 

 

 

Zbatime të derivatit të rendit II në fizikë

Le ta zemë se pika materiale kryen lëvizje drejtvizore sipas ligjit \displaystyle x=x\left( t \right). Kemi parë se shpejtësia e lëvizjes së pikës në çastin t është x’(t). Kemi pra\displaystyle V=x'\left( t \right).

Nxitimi në çastin a (që përfaqëson shpejtësinë e ndryshimit të shpejtësisë v) është sa derivati \displaystyle v'\left( a \right) i funksionit \displaystyle V=x'\left( t \right) në pikën a, dmth sa \displaystyle x''\left( a \right).

Pra, nxitimi i pikës material në lëvizjen drejtvizore është sa derivati i dytë i abshisës në lidhje me kohën.

 

Keshille! Per shembuj te zgjidhur shikoni Ushtrime te zgjidhura – Derivate

 

Ushtrime të zgjidhura

Ushtrimi 1

Gjeni derivatin e funksionit y=tgx-cotgx

 

Zgjidhje

Për të gjetur derivatin e diferencës, mjafton të gjejmë diferencën e derivateve.

\displaystyle y'=\left( tgx \right)'-\left( cotgx \right)'

\displaystyle y'=\frac{1}{{{\cos }^{2}}x}-\left( -\frac{1}{{{\sin }^{2}}x} \right)

\displaystyle y'=\frac{1}{{{\cos }^{2}}x}+\frac{1}{{{\sin }^{2}}x}

Gjejmë emëruesin e përbashkët:

\displaystyle y'=\frac{{{\sin }^{2}}x+{{\cos }^{2}}x}{{{\cos }^{2}}x\cdot {{\sin }^{2}}x}

Nga vetia themelore e trigonometrisë dimë që \displaystyle {{\sin }^{2}}x+{{\cos }^{2}}x=1. Shkruajmë:

\displaystyle y'=\frac{1}{{{\sin }^{2}}x\cdot {{\cos }^{2}}x}.

 

 

 

Ushtrimi 2

Për ç’vlera të x bëhet 0 derivati i funksionit \displaystyle y=x-tgx?

 

Zgjidhje

Gjejmë në fillimm derivatin:

\displaystyle y'=\left( x' \right)-\left( tgx \right)'

\displaystyle y'=1-\frac{1}{{{\cos }^{2}}x}, për çdo \displaystyle \cos x\ne 0.

Barazojmë derivatin me 0:

\displaystyle 1-\frac{1}{{{\cos }^{2}}x}=0

\displaystyle \frac{1}{{{\cos }^{2}}x}=1

\displaystyle {{\cos }^{2}}x=1

\displaystyle x=k\pi.

 

 

Keshille! Per shembuj te zgjidhur shikoni Ushtrime te zgjidhura – Derivate

 

Ushtrimi 3

Gjeni derivatin e dyte për funksionin f:

a) \displaystyle y=2{{x}^{5}}-3{{x}^{3}}+5{{x}^{2}}+x-7

b) \displaystyle y=\sin x-1

c) \displaystyle y=\frac{x-1}{x+1}

 

Zgjidhje

a) \displaystyle y=2{{x}^{5}}-3{{x}^{3}}+5{{x}^{2}}+x-7

Gjejmë derivatin e parë:

\displaystyle y'=10{{x}^{4}}-9{{x}^{2}}+10x+1

Gjejmë derivatin e dyte:

\displaystyle y''=40{{x}^{3}}-18x+10

\displaystyle y''=20{{x}^{3}}-9x+5

 

b) \displaystyle y=\sin x-1

Gjejmë derivatin e parë:

\displaystyle y'=\left( \sin x \right)'-\left( 1 \right)'

\displaystyle y'=\cos x

Gjejmë derivatin e dyte:

\displaystyle y''=\left( \cos x \right)'

\displaystyle y''=-\sin x

 

 

c) \displaystyle y=\frac{x-1}{x+1}

Gjejmë derivatin e parë:

\displaystyle y'=\frac{\left( x-1 \right)'\cdot \left( x+1 \right)-\left( x-1 \right)\cdot \left( x+1 \right)'}{{{\left( x+1 \right)}^{2}}}

\displaystyle y'=\frac{\left( x+1 \right)-\left( x-1 \right)}{{{\left( x+1 \right)}^{2}}}

\displaystyle y'=\frac{2}{{{\left( x+1 \right)}^{2}}}

Gjejmë derivatin e dyte:

\displaystyle y''=\frac{\left( 2 \right)'\cdot {{\left( x+1 \right)}^{2}}-2\cdot \left[ {{\left( x+1 \right)}^{2}} \right]'}{{{\left[ {{\left( x+1 \right)}^{2}} \right]}^{2}}}

\displaystyle y''=\frac{-2\cdot \left( 2x+2 \right)}{{{\left( x+1 \right)}^{4}}}

\displaystyle y''=\frac{-4x-4}{{{\left( x+1 \right)}^{4}}}

\displaystyle y''=\frac{-4\left( x+1 \right)}{{{\left( x+1 \right)}^{4}}}

\displaystyle y''=\frac{-4}{{{\left( x+1 \right)}^{3}}}

 

Keshille! Per shembuj te zgjidhur shikoni Ushtrime te zgjidhura – Derivate

 

Ushtrimi 4

Të caktohen koeficientët a,b në mënyrë që funksioni \displaystyle y=a{{x}^{2}}+bx+5 të ketë \displaystyle y'=1 dhe \displaystyle y''=1

 

Zgjidhje

Gjejmë derivatin e parë:

\displaystyle y'=2ax+b

Gjejmë derivatin e dyte:

\displaystyle y''=2a

 

Gjejmë koeficientët a dhe b:

\displaystyle y''=2a=1

\displaystyle a=\frac{1}{2}

 

\displaystyle y'=2\cdot \frac{1}{2}x+b=1

\displaystyle x+b=1

\displaystyle b=0

 

 

 

 

Ushtrimi 5

Një pikë materiale kryen lëvizje drejtvizore gjatë boshtit Ox sipas ligjit \displaystyle x=\frac{1}{3}{{t}^{3}}-2{{t}^{2}}+3t, ku x matet në metra dhe t matet në sekonda.

Në cilin cast nxitimi i pikës është zero?

 

Zgjidhje

Nga zbatimet e derivatit të rendit të dyte ne fizikë dimë që nxitimi i pikës material në lëvizjen drejtvizore është sa derivati i dytë i abshisës në lidhje me kohën.

Gjejmë në fillim derivatin e parë:

\displaystyle x'={{t}^{2}}-4t+3

Gjejmë derivatin e dytë:

\displaystyle x''=2t-4

Barazojmë derivatin e dytë me zero:

\displaystyle 2t-4=0

\displaystyle t=\frac{4}{2}=2 sekonda.

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