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Derivative=Díorthach

Finding the. Derivative=Díorthach. Cé chomh géar agus atá cnoc?. 1 ·4. 1 ·3. 0 ·3. –0 ·2. –1 ·4. 0 ·9. 1 ·0. –1 ·6. 1 ·7. 2. 0 ·8. 1. –1 ·9. 0 ·6. 0. Fána chlár scátála:. Is faide an clár…. … is lú cruinne an fána. Ag féachtaint ar phíosa beag. P.

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Derivative=Díorthach

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  1. Finding the Derivative=Díorthach

  2. Cé chomh géar agus atá cnoc? 1·4 1·3 0·3 –0·2 –1·4 0·9 1·0 –1·6 1·7 2 0·8 1 –1·9 0·6 0 Fána chlár scátála:

  3. Is faide an clár…. … is lú cruinne an fána

  4. Ag féachtaint ar phíosa beag P agus ag féachaint ar an bhfána ag pointe amháin…

  5. Gheobhfaimid luach níos cruinne… P Má bhogaimid na rothaí nios gaire le chéile

  6. Gheobhfaimid luach níos cruinne… P agus ag féachaint ar an bhfána ag pointe amháin…

  7. Sainmhíniú ar fhána: Is é fána ag pointe P ná teorainn na bhfánaí le roth ag P agus an roth eile ag dul i dtreo P. P

  8. lim y lim x→0 h→0 x f (x +h) – f (x) h Difreáil de réir na nbunphrionsabail

  9. Graf y = f(x) y=f (x) (x, y) P y x

  10. Anois tóg pointe fad ó P: y=f (x) (x +x, y +y) y (x, y) P y x x

  11. Cad é fána an líne seo? y=f (x) y2–y1 y x2–x1 x y P x

  12. y x Cad a tharlaíonn nuair a éiríonn x níos lú y P x

  13. y x Cad a tharlaíonn nuair a éiríonn x níos lú y P x

  14. y x Cad a tharlaíonn nuair a éiríonn x níos lú y P x

  15. y x Cad a tharlaíonn nuair a éiríonn x níos lú y P x

  16. y x Cad a tharlaíonn nuair a éiríonn x níos lú P y x

  17. y x Cad a tharlaíonn nuair a éiríonn x níos lú P Éiríonn an fána níos gaire fána na tadhaille

  18. lim y x→0 x dy dx Fána an tadhlaí ag an gcuar P f '(x)

  19. dy dx lim ∆x→0 ∆x ∆x ∆x ∆y ∆x = 2x Difreáil x2de réir na mbunphrionsal maidir le x :. 20 y=x2 Athraigh x go to x+Dx, agus athraigh y go y+Dy y+∆y =(x + ∆x)2 =(x + ∆x)(x + ∆x) =x2 +x∆x +x∆x +∆x2 y+ ∆y = x2+ 2x∆x +∆x2 Dealaigh y = x2 ∆y = 2x∆x + ∆x2 Roinn ar Dx Ceallaigh = 2x + ∆x Lig do Dx=0

  20. Ceisteanna Ardtéistiméirachta 2007 Q8 (b) x2 – 3x 2005 Q6 (b) 3x–x2 2004 Q8 (b) x2 +3x 2003 Q6 (b) x2 –2x 2001 Q8 (b) 3x2 –x 2000 Q6 (a) 7x+ 3

  21. dy dx lim ∆x→0 ∆x ∆x ∆x ∆x ∆y ∆x = 2x – 3 (b) Difreáil x2 -3x de réir na mbunphrionsal maidir le x . Each term inside the brackets changes sign 20 Let y=x2 – 3x Athraigh x go to x+Dx, agus athraigh y go y+Dy y+∆y = (x + ∆x)2– 3(x+ ∆x) =(x + ∆x)(x + ∆x) – 3(x + ∆x) = x2 +x∆x +x∆x +∆x2 – 3x – 3∆x y+ ∆y = x2+ 2x∆x +∆x2 – 3x – 3∆x Dealaigh y = x2 – 3x ∆y = 2x∆x + ∆x2 – 3∆x Roinn ar Dx Ceallaigh = 2x + ∆x – 3 Lig do Dx=0

  22. dy dx lim ∆x→0 ∆x ∆x ∆x ∆x ∆y ∆x = 2x+ 3 (b) Difreáil x2 +3x de réir na mbunphrionsal maidir le x . 20 Let y=x2+ 3x Athraigh x go to x+Dx, agus athraigh y go y+Dy y+∆y =(x + ∆x)2+ 3(x+ ∆x) =(x + ∆x)(x + ∆x) + 3(x + ∆x) =x2 +x∆x +x∆x +∆x2 + 3x + 3∆x y+ ∆y = x2+ 2x∆x +∆x2 + 3x + 3∆x Dealaigh y = x2 + 3x ∆y = 2x∆x + ∆x2 + 3∆x Roinn ar Dx Ceallaigh Lig do Dx=0 = 2x + ∆x +3

  23. dy dx lim ∆x→0 ∆x ∆x ∆x ∆x ∆y ∆x = 6x – 1 (b) Difreáil 3x2- -x de réir na mbunphrionsal maidir le x . Each term inside the brackets changes sign 20 Let y=3x2–x Athraigh x go to x+Dx, agus athraigh y go y+Dy y+∆y = 3(x+ ∆x)2– (x + ∆x) =3(x + ∆x)(x + ∆x) – (x + ∆x) = 3 (x2 + 2x∆x + x∆x +x∆x +∆x2) – x – ∆x y+ ∆y =3x2 + 6x∆x + 3∆x2 – x –∆x Dealaigh y = 3x2 – x 1 ∆y = 6x∆x –3∆x2 – ∆x Roinn ar Dx Ceallaigh = 6x – 3∆x – 1 Lig do Dx=0

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