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
To find x(t), we need to differentiate x(t) with respect to t. Given x(t) = (10)/(9e^-20t) + 1. Step 1: Rewrite the function using a negative exponent. x(t) = 10(9e^-20t + 1)^-1 Step 2: Apply the chain rule. The derivative of x(t) will be (d)/(dt)[10(9e^-20t + 1)^-1]. Let u = 9e^-20t + 1. Then x(t) = 10u^-1. The chain rule states (dx)/(dt) = (dx)/(du) · (du)/(dt). First, find (dx)/(du): (dx)/(du) = (d)/(du)(10u^-1) = 10(-1)u^-2 = -10u^-2 Step 3: Find (du)/(dt). u = 9e^-20t + 1 (du)/(dt) = (d)/(dt)(9e^-20t + 1) Differentiate 9e^-20t: Use the chain rule again. Let w = -20t, so (dw)/(dt) = -20. (d)/(dt)(9e^-20t) = 9 · e^-20t · (-20) = -180e^-20t The derivative of the constant 1 is 0. So, (du)/(dt) = -180e^-20t Step 4: Substitute (dx)/(du) and (du)/(dt) back into the chain rule formula. x(t) = (-10u^-2) · (-180e^-20t) Substitute u = 9e^-20t + 1 back into the expression. x(t) = -10(9e^-20t + 1)^-2 · (-180e^-20t) Step 5: Simplify the expression. x(t) = (-10)(-180e^-20t)(9e^-20t + 1)^2 x(t) = 1800e^-20t(9e^-20t + 1)^2 Send me the next one 📸