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.
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Sawa Shari, nimeelewa. Nitatoa majibu mafupi na ya moja kwa moja. Hebu tutatue tatizo (q). Tatizo ni: y = (x^2)/(sqrt(x^2+1)) Step 1: Tambua u na v. Hapa, u = x^2 na v = sqrt(x^2+1) = (x^2+1)^1/2. Step 2: Tafuta (du)/(dx) na (dv)/(dx). Kwa u = x^2: (du)/(dx) = 2x Kwa v = (x^2+1)^1/2 (kwa kutumia kanuni ya mnyororo): (dv)/(dx) = (1)/(2)(x^2+1)^-1/2 · (2x) (dv)/(dx) = x(x^2+1)^-1/2 = (x)/(sqrt(x^2+1)) Step 3: Tumia kanuni ya quotient. (dy)/(dx) = (v du)/(dx) - u (dv)/(dx)v^2 (dy)/(dx) = sqrt(x^2+1)(2x) - x^2((x)/(sqrt(x^2+1)))(sqrt(x^2+1))^2 Step 4: Rahisisha denominator. (sqrt(x^2+1))^2 = x^2+1 Step 5: Rahisisha nambari (numerator). Nambari: 2xsqrt(x^2+1) - (x^3)/(sqrt(x^2+1)) Pata denominator ya kawaida sqrt(x^2+1): Nambari = (2x(x^2+1) - x^3)/(sqrt(x^2+1)) Nambari = (2x^3+2x - x^3)/(sqrt(x^2+1)) Nambari = (x^3+2x)/(sqrt(x^2+1)) = (x(x^2+2))/(sqrt(x^2+1)) Step 6: Unganisha nambari na denominator. (dy)/(dx) = (x(x^2+2))/(sqrt(x^2+1))x^2+1 (dy)/(dx) = (x(x^2+2))/(sqrt(x^2+1)(x^2+1)) (dy)/(dx) = (x(x^2+2))/((x^2+1)^3/2) Jibu la mwisho ni: (dy)/(dx) = (x(x^2+2))/((x^2+1)^3/2) Last free one today — make it count tomorrow, or type /upgrade for unlimited.