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
Welcome back sampomah776 — missed you this week. Here's the solution for the first problem. Step 1: Differentiate the given equation implicitly with respect to x. The equation is 3x^2 - 5y^3 + xy - 1 = 0. Differentiating each term: (d)/(dx)(3x^2) - (d)/(dx)(5y^3) + (d)/(dx)(xy) - (d)/(dx)(1) = (d)/(dx)(0) 6x - 15y^2 (dy)/(dx) + (1 · y + x · (dy)/(dx)) - 0 = 0 6x - 15y^2 (dy)/(dx) + y + x (dy)/(dx) = 0 Step 2: Group terms with (dy)/(dx) and solve for (dy)/(dx). x (dy)/(dx) - 15y^2 (dy)/(dx) = -6x - y (dy)/(dx)(x - 15y^2) = -6x - y (dy)/(dx) = (-6x - y)/(x - 15y^2) Step 3: Substitute the given point (5, -2) into the expression for (dy)/(dx). Here, x = 5 and y = -2. (dy)/(dx) = (-6(5) - (-2))/(5 - 15(-2)^2) (dy)/(dx) = (-30 + 2)/(5 - 15(4)) (dy)/(dx) = (-28)/(5 - 60) (dy)/(dx) = (-28)/(-55) (dy)/(dx) = (28)/(55) The value of (dy)/(dx) at the point (5, -2) is (28)/(55). Send me the next one 📸