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: Rewrite the function for easier differentiation. The given function is y = (20)/(x^2+1). This can be written as y = 20(x^2+1)^-1. Step 2: Differentiate the function with respect to x. We use the chain rule: (dy)/(dx) = (d)/(dx) [20(x^2+1)^-1]. (dy)/(dx) = 20 · (-1)(x^2+1)^-2 · (d)/(dx)(x^2+1) (dy)/(dx) = -20(x^2+1)^-2 · (2x) (dy)/(dx) = -(40x)/((x^2+1)^2) Step 3: Substitute the x-coordinate of the given point into the derivative. The given point is (3,2), so x=3. (dy)/(dx)|_x=3 = -(40(3))/(((3)^2+1)^2) (dy)/(dx)|_x=3 = -(120)/((9+1)^2) (dy)/(dx)|_x=3 = -(120)/((10)^2) (dy)/(dx)|_x=3 = -(120)/(100) Step 4: Simplify the result. (dy)/(dx)|_x=3 = -(12)/(10) = -(6)/(5) The gradient of the curve at the point (3,2) is -(6)/(5). Send me the next one 📸