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 differentiations for each function: To differentiate, we use the power rule: (d)/(dx)(ax^n) = anx^n-1 and the rule for constants: (d)/(dx)(c) = 0. i) y = 3x^7 + 2x^3 - 3x Step 1: Differentiate each term with respect to x. (dy)/(dx) = (d)/(dx)(3x^7) + (d)/(dx)(2x^3) - (d)/(dx)(3x) Step 2: Apply the power rule to each term. (dy)/(dx) = (3 · 7x^7-1) + (2 · 3x^3-1) - (3 · 1x^1-1) (dy)/(dx) = 21x^6 + 6x^2 - 3x^0 Step 3: Simplify the expression (x^0 = 1). (dy)/(dx) = 21x^6 + 6x^2 - 3 ii) f(x) = 4x^9 - 5x^10 + 17 Step 1: Differentiate each term with respect to x. f'(x) = (d)/(dx)(4x^9) - (d)/(dx)(5x^10) + (d)/(dx)(17) Step 2: Apply the power rule and the constant rule. f'(x) = (4 · 9x^9-1) - (5 · 10x^10-1) + 0 f'(x) = 36x^8 - 50x^9 Step 3: Simplify the expression. f'(x) = 36x^8 - 50x^9 iii) g(x) = x^3 - 4x^5 + 16x - 3 Step 1: Differentiate each term with respect to x. g'(x) = (d)/(dx)(x^3) - (d)/(dx)(4x^5) + (d)/(dx)(16x) - (d)/(dx)(3) Step 2: Apply the power rule and the constant rule. g'(x) = (1 · 3x^3-1) - (4 · 5x^5-1) + (16 · 1x^1-1) - 0 g'(x) = 3x^2 - 20x^4 + 16x^0 Step 3: Simplify the expression (x^0 = 1). g'(x) = 3x^2 - 20x^4 + 16 That's 2 down. 3 left today — send the next one.