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, hebu tuendelee na tatizo (n). Ili kutatua tatizo (n), tutatumia kanuni ya mnyororo na kanuni ya nguvu. Tatizo ni: y = (sqrt(x) - (1)/(x))^2 Step 1: Andika upya usemi kwa kutumia viashiria vya nguvu. y = (x^1/2 - x^-1)^2 Step 2: Tumia kanuni ya mnyororo. Acha u = x^1/2 - x^-1. Kisha y = u^2. Tafuta (du)/(dx): (du)/(dx) = (d)/(dx)(x^1/2) - (d)/(dx)(x^-1) (du)/(dx) = (1)/(2)x^1/2 - 1 - (-1)x^-1-1 (du)/(dx) = (1)/(2)x^-1/2 + x^-2 Tafuta (dy)/(du): (dy)/(du) = (d)/(du)(u^2) = 2u Step 3: Tumia kanuni ya mnyororo (dy)/(dx) = (dy)/(du) · (du)/(dx). Badilisha u na (du)/(dx): (dy)/(dx) = 2(x^1/2 - x^-1) ((1)/(2)x^-1/2 + x^-2) Step 4: Rahisisha usemi. Panua maneno: (dy)/(dx) = 2(x^1/2 · (1)/(2)x^-1/2 + x^1/2 · x^-2 - x^-1 · (1)/(2)x^-1/2 - x^-1 · x^-2) (dy)/(dx) = 2((1)/(2)x^1/2 - 1/2 + x^1/2 - 2 - (1)/(2)x^-1 - 1/2 - x^-1 - 2) (dy)/(dx) = 2((1)/(2)x^0 + x^-3/2 - (1)/(2)x^-3/2 - x^-3) (dy)/(dx) = 2((1)/(2) + x^-3/2 - (1)/(2)x^-3/2 - x^-3) Changanya maneno yanayofanana: (dy)/(dx) = 2((1)/(2) + (1)/(2)x^-3/2 - x^-3) Gawanya kwa 2: (dy)/(dx) = 1 + x^-3/2 - 2x^-3 Step 5: Andika jibu kwa kutumia mizizi na sehemu. (dy)/(dx) = 1 + (1)/(x^3/2) - (2)/(x^3) (dy)/(dx) = 1 + (1)/(xsqrt(x)) - (2)/(x^3) Jibu la mwisho ni: (dy)/(dx) = 1 + (1)/(xsqrt(x)) - (2)/(x^3) That's 2 down. 3 left today — send the next one.