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
Okay, Ni_Fe_Mi✨. Let's differentiate this one. The problem is to differentiate y = (3sqrt(x) - 2x)^3. Step 1: Rewrite the function using fractional exponents for the inner term. y = (3x^1/2 - 2x)^3 Step 2: Apply the chain rule. Let u = 3x^1/2 - 2x. The derivative of y = u^3 is (dy)/(dx) = 3u^2 · (du)/(dx). (dy)/(dx) = 3(3x^1/2 - 2x)^3-1 · (d)/(dx)(3x^1/2 - 2x) Step 3: Differentiate the inner function (d)/(dx)(3x^1/2 - 2x). (d)/(dx)(3x^1/2 - 2x) = 3 · (1)/(2)x^(1)/(2)-1 - 2 · 1 = (3)/(2)x^-1/2 - 2 Step 4: Substitute the derivative of the inner function back into the expression from Step 2 and simplify. (dy)/(dx) = 3(3x^1/2 - 2x)^2 ((3)/(2)x^-1/2 - 2) Step 5: Rewrite the terms with positive and radical exponents for clarity. (dy)/(dx) = 3(3sqrt(x) - 2x)^2 ((3)/(2sqrt(x)) - 2) The derivative is 3(3sqrt(x) - 2x)^2 ((3)/(2sqrt(x)) - 2). 3 done, 2 left today. You're making progress.