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 the first and second derivatives of the given function Z = 4x^3 - 5x^2 + 3x, we will use the power rule for differentiation, which states that (d)/(dx)(ax^n) = nax^n-1. Step 1: Find the first derivative, (dZ)/(dx). We differentiate each term of the function with respect to x: (dZ)/(dx) = (d)/(dx)(4x^3) - (d)/(dx)(5x^2) + (d)/(dx)(3x) Applying the power rule: (dZ)/(dx) = (3 · 4x^3-1) - (2 · 5x^2-1) + (1 · 3x^1-1) (dZ)/(dx) = 12x^2 - 10x^1 + 3x^0 (dZ)/(dx) = 12x^2 - 10x + 3 Step 2: Find the second derivative, (d^2Z)/(dx^2). We differentiate the first derivative, (dZ)/(dx), with respect to x: (d^2Z)/(dx^2) = (d)/(dx)(12x^2 - 10x + 3) Differentiating each term: (d^2Z)/(dx^2) = (d)/(dx)(12x^2) - (d)/(dx)(10x) + (d)/(dx)(3) Applying the power rule: (d^2Z)/(dx^2) = (2 · 12x^2-1) - (1 · 10x^1-1) + 0 (d^2Z)/(dx^2) = 24x^1 - 10x^0 (d^2Z)/(dx^2) = 24x - 10 The first derivative is: (dZ)/(dx) = 12x^2 - 10x + 3 The second derivative is: (d^2Z)/(dx^2) = 24x - 10 That's 2 down. 3 left today — send the next one.