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
Step 1: Andika upya kazi z kwa fomu rahisi ya kutofautisha. Kazi uliyopewa ni z = (1)/(sqrt(x^2 + y^2)). Hii inaweza kuandikwa kama: z = (x^2 + y^2)^-1/2 Step 2: Pata (dz)/(dx) (derivative ya sehemu ya z kwa heshima ya x). Ili kupata (dz)/(dx), tunachukulia y kama kigezo kisichobadilika (constant). Tutatumia chain rule. Kanuni ya chain rule ni (d)/(dx)[u^n] = n u^n-1 (du)/(dx). Hapa, u = x^2 + y^2 na n = -(1)/(2). Kwanza, pata (du)/(dx): (du)/(dx) = (d)/(dx)(x^2 + y^2) = 2x + 0 = 2x Sasa, tumia chain rule: (dz)/(dx) = -(1)/(2)(x^2 + y^2)^-1/2 - 1 · (2x) (dz)/(dx) = -(1)/(2)(x^2 + y^2)^-3/2 · (2x) Rahisisha usemi: (dz)/(dx) = -x(x^2 + y^2)^-3/2 Au kwa fomu ya mizizi: (dz)/(dx) = -(x)/((x^2 + y^2)^3/2) Step 3: Pata (dz)/(dy) (derivative ya sehemu ya z kwa heshima ya y). Ili kupata (dz)/(dy), tunachukulia x kama kigezo kisichobadilika (constant). Tutatumia chain rule tena. Hapa, u = x^2 + y^2 na n = -(1)/(2). Kwanza, pata (du)/(dy): (du)/(dy) = (d)/(dy)(x^2 + y^2) = 0 + 2y = 2y Sasa, tumia chain rule: (dz)/(dy) = -(1)/(2)(x^2 + y^2)^-1/2 - 1 · (2y) (dz)/(dy) = -(1)/(2)(x^2 + y^2)^-3/2 · (2y) Rahisisha usemi: (dz)/(dy) = -y(x^2 + y^2)^-3/2 Au kwa fomu ya mizizi: (dz)/(dy) = -(y)/((x^2 + y^2)^3/2) Majibu ni: (dz)/(dx) = -(x)/((x^2 + y^2)^3/2) (dz)/(dy) = -(y)/((x^2 + y^2)^3/2) That's 2 down. 3 left today — send the next one.