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.
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Here are the solutions for the differentiation questions from the image. 4. Differentiate the following The general power rule for differentiation is (d)/(dx)(ax^n) = anx^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 to each term. (dy)/(dx) = 7(4x^4-1) + 5(3x^3-1) - 7(1x^1-1) (dy)/(dx) = 28x^3 + 15x^2 - 7x^0 (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^3 - x + 5x Step 1: Simplify the expression for y by combining like terms. y = 8x^3 + (-x + 5x) y = 8x^3 + 4x Step 2: Differentiate each term with respect to x. (dy)/(dx) = (d)/(dx)(8x^3) + (d)/(dx)(4x) Step 3: Apply the power rule to each term. (dy)/(dx) = 8(3x^3-1) + 4(1x^1-1) (dy)/(dx) = 24x^2 + 4x^0 (dy)/(dx) = 24x^2 + 4 The final answer for 4(3) is 24x^2 + 4. (4) y = x^-7 Step 1: Differentiate y with respect to x using the power rule. (dy)/(dx) = -7x^-7-1 (dy)/(dx) = -7x^-8 The final answer for 4(4) is -7x^-8.