Kopertina e librit Matematika 12

Zgjidhja e ushtrimit 1

Zgjidhja e ushtrimit 1 të mësimit 4.2A në librin Matematika 12 nga shtëpia botuese Botime Pegi me autorë Brian Jefferson, David Bowles, Eddie Mullan, Garry Wiseman, John Rayneau, Katie Wood, Mike Heylings, Paul Williams dhe Rob Wagner.

Pyetja

Gjeni derivatet e funksioneve të mëposhtme y=f(x)y=f(x), ku f(x)f(x) është:

  1. 3x73x^7
  2. 6x46x^4
  3. 5x5x
  4. 88
  5. −4-4
  6. 13\dfrac{1}{3}
  7. −2x5-2x^5
  8. −x7-x^7
  9. 2x−12x^{-1}
  10. −x−3-x^{-3}
  11. x12x^{\frac{1}{2}}
  12. x23x^{\frac{2}{3}}
  13. 5x535x^{\frac{5}{3}}
  14. −x−14-x^{-\frac{1}{4}}
  15. 00

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Zgjidhja

  1. dydx\dfrac{\text{d}y}{\text{d}x} == 7×3x7−1=21x67 \times 3x^{7-1} = 21x^6
  2. dydx=4×6x4−1=24x3\dfrac{\text{d}y}{\text{d}x} = 4 \times 6x^{4-1} = 24x^3
  3. dydx=5\dfrac{\text{d}y}{\text{d}x} = 5
  4. dydx=0\dfrac{\text{d}y}{\text{d}x} = 0
  5. dydx=0\dfrac{\text{d}y}{\text{d}x} = 0
  6. dydx=0\dfrac{\text{d}y}{\text{d}x} = 0
  7. dydx=5×−2x5−1=−10x4\dfrac{\text{d}y}{\text{d}x} = 5 \times -2x^{5-1} = -10x^4
  8. dydx=7×−x7−1=−7x6\dfrac{\text{d}y}{\text{d}x} = 7 \times -x^{7-1} = -7x^6
  9. dydx=−1×2x−1−1=−2x−2\dfrac{\text{d}y}{\text{d}x} = -1 \times 2x^{-1-1} = -2x^{-2}
  10. dydx=−3×−x−3−1=3x−4\dfrac{\text{d}y}{\text{d}x} = -3 \times -x^{-3-1} = 3x^{-4}
  11. dydx=12×x12−1=12x−12\dfrac{\text{d}y}{\text{d}x} = \dfrac{1}{2} \times x^{\frac{1}{2}-1} = \dfrac{1}{2}x^{-\frac{1}{2}}
  12. dydx=23×x23−1=23x−13\dfrac{\text{d}y}{\text{d}x} = \dfrac{2}{3} \times x^{\frac{2}{3}-1} = \dfrac{2}{3}x^{-\frac{1}{3}}
  13. dydx=53×5x53−1=253x23\dfrac{\text{d}y}{\text{d}x} = \dfrac{5}{3} \times 5x^{\frac{5}{3}-1} = \dfrac{25}{3}x^{\frac{2}{3}}
  14. dydx=−14×−x−14−1=14x−54\dfrac{\text{d}y}{\text{d}x} = -\dfrac{1}{4} \times -x^{-\frac{1}{4}-1} = \dfrac{1}{4}x^{-\frac{5}{4}}
  15. dydx=0\dfrac{\text{d}y}{\text{d}x} = 0