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
Here are the solutions for the remaining differentiation questions from the image. 4. Differentiate the following The general power rule for differentiation is (d)/(dx)(x^n) = nx^n-1. (1) y = 7x^4 + 5x^3 - 7x Step 1: Differentiate each term with respect to x. (dy)/(dx) = (d)/(dx)(7x^4) + (d)/(dx)(5x^3) - (d)/(dx)(7x) Step 2: Apply the power rule and constant multiple rule. (dy)/(dx) = 7(4x^4-1) + 5(3x^3-1) - 7(1x^1-1) (dy)/(dx) = 7(4x^3) + 5(3x^2) - 7(1x^0) (dy)/(dx) = 28x^3 + 15x^2 - 7(1) (dy)/(dx) = 28x^3 + 15x^2 - 7 The final answer for 4(1) is 28x^3 + 15x^2 - 7. (2) y = x^3 Step 1: Differentiate y with respect to x using the power rule. (dy)/(dx) = 3x^3-1 (dy)/(dx) = 3x^2 The final answer for 4(2) is 3x^2. (3) y = 8x^2 - x + 5x Step 1: Simplify the expression for y first. y = 8x^2 + (-x + 5x) y = 8x^2 + 4x Step 2: Differentiate each term with respect to x. (dy)/(dx) = (d)/(dx)(8x^2) + (d)/(dx)(4x) Step 3: Apply the power rule and constant multiple rule. (dy)/(dx) = 8(2x^2-1) + 4(1x^1-1) (dy)/(dx) = 8(2x^1) + 4(1x^0) (dy)/(dx) = 16x + 4(1) (dy)/(dx) = 16x + 4 The final answer for 4(3) is 16x + 4. (4) y = x^n Step 1: Differentiate y with respect to x using the power rule. (dy)/(dx) = nx^n-1 The final answer for 4(4) is nx^n-1.