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 is the solution for question 4 from the image. 4. Differentiate the following (1) y = 7x^4 + 5x^3 - 7x Step 1: Apply the power rule for differentiation, (d)/(dx)(ax^n) = anx^n-1, and the sum/difference rule. (dy)/(dx) = (d)/(dx)(7x^4) + (d)/(dx)(5x^3) - (d)/(dx)(7x) Step 2: Differentiate each term. (d)/(dx)(7x^4) = 7 · 4x^4-1 = 28x^3 (d)/(dx)(5x^3) = 5 · 3x^3-1 = 15x^2 (d)/(dx)(7x) = 7 · 1x^1-1 = 7x^0 = 7 Step 3: Combine the derivatives. (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: Apply the power rule for differentiation, (d)/(dx)(x^n) = nx^n-1. (dy)/(dx) = 3x^3-1 Step 2: Simplify the exponent. (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. y = 8x^2 + (-x + 5x) y = 8x^2 + 4x Step 2: Apply the power rule for differentiation, (d)/(dx)(ax^n) = anx^n-1, and the sum rule. (dy)/(dx) = (d)/(dx)(8x^2) + (d)/(dx)(4x) Step 3: Differentiate each term. (d)/(dx)(8x^2) = 8 · 2x^2-1 = 16x^1 = 16x (d)/(dx)(4x) = 4 · 1x^1-1 = 4x^0 = 4 Step 4: Combine the derivatives. (dy)/(dx) = 16x + 4 The final answer for 4(3) is 16x + 4. (4) y = x^n Step 1: Apply the power rule for differentiation directly. (dy)/(dx) = nx^n-1 The final answer for 4(4) is nx^n-1.