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 — missed you this week. To find (dy)/(dx) at the point (1,1) for the implicitly defined function x^y(x) + y(x)x = (xy(x)), we will use implicit differentiation. Step 1: Differentiate each term of the equation with respect to x. The given equation is: x^y + yx = (xy) We need to find the derivative of each term. For x^y, we use logarithmic differentiation or the fact that x^y = e^y x: (d)/(dx)(x^y) = (d)/(dx)(e^y x) = e^y x · (d)/(dx)(y x) = x^y ( (dy)/(dx) x + y · (1)/(x) ) For yx, we use the product rule: (d)/(dx)(yx) = (dy)/(dx) · x + y · 1 = x (dy)/(dx) + y For (xy), we use the chain rule and product rule: (d)/(dx)((xy)) = (xy) · (d)/(dx)(xy) = (xy) ( (dy)/(dx) · x + y · 1 ) = (xy) ( x (dy)/(dx) + y ) Step 2: Substitute the derivatives back into the equation. x^y ( (dy)/(dx) x + (y)/(x) ) + x (dy)/(dx) + y = (xy) ( x (dy)/(dx) + y ) Step 3: Substitute the point (x,y) = (1,1) into the differentiated equation. Recall that (1) = 0. 1^1 ( (dy)/(dx) 1 + (1)/(1) ) + 1 · (dy)/(dx) + 1 = (1 · 1) ( 1 · (dy)/(dx) + 1 ) 1 ( (dy)/(dx) · 0 + 1 ) + (dy)/(dx) + 1 = (1) ( (dy)/(dx) + 1 ) Step 4: Simplify the equation. 1 + (dy)/(dx) + 1 = (1) (dy)/(dx) + (1) 2 + (dy)/(dx) = (1) (dy)/(dx) + (1) Step 5: Isolate (dy)/(dx). Subtract (1) (dy)/(dx) from both sides and subtract 2 from both sides: (dy)/(dx) - (1) (dy)/(dx) = (1) - 2 Factor out (dy)/(dx): (dy)/(dx) (1 - (1)) = (1) - 2 Divide by (1 - (1)): (dy)/(dx) = ((1) - 2)/(1 - (1)) The value of (dy)/(dx) at the point (1,1) is: ((1) - 2)/(1 - (1)) Send me the next one 📸