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
Morning Theamazingdivine — let's get this done. The problem asks to "Solve this", which for a complex algebraic expression typically means finding its derivative. We will use logarithmic differentiation to find the derivative of the given function. Let y be the given expression: y = ((6x^2+1)^3 (2x-5)^4)/((x^3-2x+1)^2) Step 1: Take the natural logarithm of both sides. y = ( ((6x^2+1)^3 (2x-5)^4)/((x^3-2x+1)^2) ) Step 2: Use logarithm properties ((AB) = A + B, ((A)/(B)) = A - B, (A^n) = n A) to expand the right side. y = 3 (6x^2+1) + 4 (2x-5) - 2 (x^3-2x+1) Step 3: Differentiate both sides with respect to x. Remember that (d)/(dx)( u) = (1)/(u) (du)/(dx). (1)/(y) (dy)/(dx) = 3 · (1)/(6x^2+1) · (d)/(dx)(6x^2+1) + 4 · (1)/(2x-5) · (d)/(dx)(2x-5) - 2 · (1)/(x^3-2x+1) · (d)/(dx)(x^3-2x+1) Step 4: Calculate the derivatives of the inner functions and substitute them back. (d)/(dx)(6x^2+1) = 12x (d)/(dx)(2x-5) = 2 (d)/(dx)(x^3-2x+1) = 3x^2-2 Substituting these into the equation from Step 3: (1)/(y) (dy)/(dx) = 3 · (12x)/(6x^2+1) + 4 · (2)/(2x-5) - 2 · (3x^2-2)/(x^3-2x+1) (1)/(y) (dy)/(dx) = (36x)/(6x^2+1) + (8)/(2x-5) - (2(3x^2-2))/(x^3-2x+1) Step 5: Multiply both sides by y to solve for (dy)/(dx). (dy)/(dx) = y ( (36x)/(6x^2+1) + (8)/(2x-5) - (2(3x^2-2))/(x^3-2x+1) ) Step 6: Substitute the original expression for y back into the equation. (dy)/(dx) = ((6x^2+1)^3 (2x-5)^4)/((x^3-2x+1)^2) ( (36x)/(6x^2+1) + (8)/(2x-5) - (2(3x^2-2))/(x^3-2x+1) ) The final answer is: (dy)/(dx) = ((6x^2+1)^3 (2x-5)^4)/((x^3-2x+1)^2) ( (36x)/(6x^2+1) + (8)/(2x-5) - (2(3x^2-2))/(x^3-2x+1) ) What's next? 🚀