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: The given function is: y = 2x^5 + 4x^4 - x^3 + 3x^2 - 5x + 7 Step 2: Differentiate each term of the polynomial with respect to x using the power rule (d)/(dx)(ax^n) = anx^n-1 and the rule that the derivative of a constant is 0. (dy)/(dx) = (d)/(dx)(2x^5) + (d)/(dx)(4x^4) - (d)/(dx)(x^3) + (d)/(dx)(3x^2) - (d)/(dx)(5x) + (d)/(dx)(7) (dy)/(dx) = (2 · 5)x^5-1 + (4 · 4)x^4-1 - (1 · 3)x^3-1 + (3 · 2)x^2-1 - (5 · 1)x^1-1 + 0 Step 3: Simplify the expression for (dy)/(dx). (dy)/(dx) = 10x^4 + 16x^3 - 3x^2 + 6x - 5 Step 4: Substitute x=2 into the expression for (dy)/(dx) to find its value. (dy)/(dx) |_x=2 = 10(2)^4 + 16(2)^3 - 3(2)^2 + 6(2) - 5 (dy)/(dx) |_x=2 = 10(16) + 16(8) - 3(4) + 12 - 5 (dy)/(dx) |_x=2 = 160 + 128 - 12 + 12 - 5 Step 5: Perform the arithmetic operations. (dy)/(dx) |_x=2 = 288 - 12 + 12 - 5 (dy)/(dx) |_x=2 = 276 + 12 - 5 (dy)/(dx) |_x=2 = 288 - 5 (dy)/(dx) |_x=2 = 283 That's 2 down. 3 left today — send the next one.