1
Hosila jadvali va hosila qoidalari
maktab darsligi: matematika 11-sinf, hosila va uning hisoblash qoidalari
Maqsad: Elementar funksiyalar hosilalarini jadvaldan va yig‘indi, ko‘paytma, bo‘linma qoidalaridan foydalanib topish.
Qoida va g‘oya
Funksiyaning x0 nuqtadagi hosilasi — orttirmalar nisbatining limiti: f'(x0) = lim (f(x0 + Δx) - f(x0))/Δx, Δx → 0. Jadval: C' = 0; (x^n)' = n · x^(n-1); (sqrt(x))' = 1/(2 sqrt(x)); (1/x)' = -1/x^2; (sin x)' = cos x; (cos x)' = -sin x; (tg x)' = 1/cos^2 x; (e^x)' = e^x; (a^x)' = a^x · ln a; (ln x)' = 1/x; (log_a x)' = 1/(x ln a). Qoidalar: (u ± v)' = u' ± v'; (Cu)' = C · u'; (uv)' = u'v + uv'; (u/v)' = (u'v - uv')/v^2. Yig‘indining hosilasi har bir hadning hosilasi yig‘indisiga teng, lekin ko‘paytmaning hosilasi hosilalar ko‘paytmasi emas.
Namunaviy masalalar
f(x) = x^3 - 4x + 5 bo‘lsa, f'(2) ni toping.
Yechim: f'(x) = 3x^2 - 4. f'(2) = 12 - 4 = 8.
Javob: 8
Yechim: f'(x) = 3x^2 - 4. f'(2) = 12 - 4 = 8.
Javob: 8
f(x) = x^2 · e^x ning hosilasini toping.
Yechim: (uv)' = 2x · e^x + x^2 · e^x = e^x (x^2 + 2x).
Javob: e^x (x^2 + 2x)
Yechim: (uv)' = 2x · e^x + x^2 · e^x = e^x (x^2 + 2x).
Javob: e^x (x^2 + 2x)
f(x) = x/(x + 1) bo‘lsa, f'(1) ni toping.
Yechim: f' = (1 · (x + 1) - x · 1)/(x + 1)^2 = 1/(x + 1)^2. f'(1) = 1/4.
Javob: 1/4
Yechim: f' = (1 · (x + 1) - x · 1)/(x + 1)^2 = 1/(x + 1)^2. f'(1) = 1/4.
Javob: 1/4
Tipik xatolar
- Ko‘paytmaning hosilasini (uv)' = u'v' deb yozish.
- (x^n)' = n · x^(n-1) da ko‘rsatkichni 1 ga kamaytirmaslik.
- Bo‘linma hosilasida suratdagi ayirma tartibini almashtirish: u'v - uv'.
Mashq
1
f(x) = 2x^4 - 3x^2 + x bo‘lsa, f'(1) ni toping.
3
2
f(x) = sqrt(x) + 1/x bo‘lsa, f'(4) ni toping.
3/16
3
f(x) = x ln x bo‘lsa, f'(e) ni toping.
2
4
f(x) = sin x + 3 cos x bo‘lsa, f'(π/2) ni toping.
-3