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
Right Hurry, let's go. Here's how to find the derivative of the given expression: We need to find the derivative of y = (x^2-3x+5)(2x-7) with respect to x. We will use the product rule, which states that if y = uv, then (dy)/(dx) = u(dv)/(dx) + v(du)/(dx). Step 1: Identify u and v and find their derivatives. Let u = x^2-3x+5. Then, the derivative of u with respect to x is: (du)/(dx) = (d)/(dx)(x^2-3x+5) = 2x-3 Let v = 2x-7. Then, the derivative of v with respect to x is: (dv)/(dx) = (d)/(dx)(2x-7) = 2 Step 2: Apply the product rule formula. (dy)/(dx) = u(dv)/(dx) + v(du)/(dx) Substitute the expressions for u, v, (du)/(dx), and (dv)/(dx): (dy)/(dx) = (x^2-3x+5)(2) + (2x-7)(2x-3) Step 3: Expand and simplify the expression. First, expand the first term: 2(x^2-3x+5) = 2x^2 - 6x + 10 Next, expand the second term: (2x-7)(2x-3) = 2x(2x-3) - 7(2x-3) = 4x^2 - 6x - 14x + 21 = 4x^2 - 20x + 21 Step 4: Combine the expanded terms. (dy)/(dx) = (2x^2 - 6x + 10) + (4x^2 - 20x + 21) Combine like terms: (dy)/(dx) = (2x^2 + 4x^2) + (-6x - 20x) + (10 + 21) (dy)/(dx) = 6x^2 - 26x + 31 The derivative of (x^2-3x+5)(2x-7) is 6x^2 - 26x + 31. What's next?