This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Ili kutatua tatizo (m), tutatumia kanuni ya quotient na kanuni ya mnyororo. Tatizo ni: y = sqrt(x+1)x-3 Step 1: Tambua u na v. Hapa, u = sqrt(x+1) = (x+1)^1/2 na v = x-3. Step 2: Tafuta (du)/(dx) na (dv)/(dx). Kwa u = (x+1)^1/2: (du)/(dx) = (1)/(2)(x+1)^(1)/(2)-1 · (d)/(dx)(x+1) (du)/(dx) = (1)/(2)(x+1)^-1/2 · 1 (du)/(dx) = (1)/(2sqrt(x+1)) Kwa v = x-3: (dv)/(dx) = 1 Step 3: Tumia kanuni ya quotient. Kanuni ya quotient inasema: (dy)/(dx) = (v du)/(dx) - u (dv)/(dx)v^2. Badilisha maadili ya u, v, (du)/(dx), (dv)/(dx): (dy)/(dx) = ((x-3)(1)/(2sqrt(x+1))) - (sqrt(x+1))(1)(x-3)^2 Step 4: Rahisisha nambari (numerator). Nambari ni: (x-3)/(2sqrt(x+1)) - sqrt(x+1) Ili kuchanganya maneno haya, tafuta denominator ya kawaida, ambayo ni 2sqrt(x+1): Nambari = (x-3)/(2sqrt(x+1)) - sqrt(x+1) · 2sqrt(x+1)2sqrt(x+1) Nambari = (x-3 - 2(x+1))/(2sqrt(x+1)) Nambari = (x-3 - 2x - 2)/(2sqrt(x+1)) Nambari = (-x-5)/(2sqrt(x+1)) Step 5: Badilisha nambari iliyorahisishwa kwenye usemi wa (dy)/(dx). (dy)/(dx) = (-x-5)/(2sqrt(x+1))(x-3)^2 (dy)/(dx) = (-x-5)/(2sqrt(x+1)(x-3)^2) Jibu la mwisho ni: (dy)/(dx) = (-x-5)/(2sqrt(x+1)(x-3)^2) Tuma swali linalofuata 📸