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 derivatives for each function: a) y = x Step 1: Apply the power rule (d)/(dx)(x^n) = nx^n-1. (dy)/(dx) = 1 · x^1-1 (dy)/(dx) = 1 · x^0 (dy)/(dx) = 1 · 1 (dy)/(dx) = 1 b) y = -x^2 + 3 Step 1: Differentiate each term using the power rule and the constant rule. (dy)/(dx) = (d)/(dx)(-x^2) + (d)/(dx)(3) (dy)/(dx) = -2x^2-1 + 0 (dy)/(dx) = -2x c) y = (x^3)/(3) + (x^2)/(2) + x Step 1: Differentiate each term using the power rule. (dy)/(dx) = (1)/(3) · (d)/(dx)(x^3) + (1)/(2) · (d)/(dx)(x^2) + (d)/(dx)(x) (dy)/(dx) = (1)/(3) · 3x^3-1 + (1)/(2) · 2x^2-1 + 1 (dy)/(dx) = x^2 + x + 1 (dy)/(dx) = x^2 + x + 1 d) y = 3x^7 - 7x^3 + 21x^2 Step 1: Differentiate each term using the power rule. (dy)/(dx) = 3 · (d)/(dx)(x^7) - 7 · (d)/(dx)(x^3) + 21 · (d)/(dx)(x^2) (dy)/(dx) = 3 · 7x^7-1 - 7 · 3x^3-1 + 21 · 2x^2-1 (dy)/(dx) = 21x^6 - 21x^2 + 42x (dy)/(dx) = 21x^6 - 21x^2 + 42x e) y = (x^4)/(4) - (x^3)/(3) + (x^2)/(2) - x + 3 Step 1: Differentiate each term using the power rule and the constant rule. (dy)/(dx) = (1)/(4) · (d)/(dx)(x^4) - (1)/(3) · (d)/(dx)(x^3) + (1)/(2) · (d)/(dx)(x^2) - (d)/(dx)(x) + (d)/(dx)(3) (dy)/(dx) = (1)/(4) · 4x^4-1 - (1)/(3) · 3x^3-1 + (1)/(2) · 2x^2-1 - 1 + 0 (dy)/(dx) = x^3 - x^2 + x - 1 (dy)/(dx) = x^3 - x^2 + x - 1 f) y = sqrt(x^5) Step 1: Rewrite the function using fractional exponents. y = x^5/2 Step 2: Apply the power rule. (dy)/(dx) = (5)/(2) x^(5)/(2) - 1 (dy)/(dx) = (5)/(2) x^(5)/(2) - (2)/(2) (dy)/(dx) = (5)/(2) x^3/2 (dy)/(dx) = (5)/(2) x^3/2 g) y = (1)/(x^2/5) Step 1: Rewrite the function using negative exponents. y = x^-2/5 Step 2: Apply the power rule. (dy)/(dx) = -(2)/(5) x^-(2)/(5) - 1 (dy)/(dx) = -(2)/(5) x^-(2)/(5) - (5)/(5) (dy)/(dx) = -(2)/(5) x^-7/5 (dy)/(dx) = -(2)/(5) x^-7/5 That's 2 down. 3 left today — send the next one.