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: 1) Given y = (2)/(sqrt(x)) + [3]x^2 - (1)/(x) Step 1: Rewrite the terms using exponent notation. y = 2x^-1/2 + x^2/3 - x^-1 Step 2: Differentiate each term using the power rule (d)/(dx)(ax^n) = anx^n-1. (dy)/(dx) = 2(-(1)/(2))x^-1/2 - 1 + (2)/(3)x^2/3 - 1 - (-1)x^-1 - 1 (dy)/(dx) = -x^-3/2 + (2)/(3)x^-1/3 + x^-2 Step 3: Rewrite the terms with positive exponents and radical notation. (dy)/(dx) = -(1)/(x^3/2) + (2)/(3x^1/3) + (1)/(x^2) (dy)/(dx) = -(1)/(xsqrt(x)) + (2)/(3[3]x) + (1)/(x^2) The final answer is -(1)/(xsqrt(x)) + (2)/(3[3]x) + (1)/(x^2). 2) Given y = 3x^4 + (2xp)^0 - 3q + (16)/([4]x^3) Step 1: Simplify the terms. (2xp)^0 = 1 (assuming 2xp ≠ 0), 3q is a constant, and (16)/([4]x^3) = 16x^-3/4. y = 3x^4 + 1 - 3q + 16x^-3/4 Step 2: Differentiate each term. The derivative of a constant is 0. (dy)/(dx) = 3(4)x^4-1 + 0 - 0 + 16(-(3)/(4))x^-3/4 - 1 (dy)/(dx) = 12x^3 - 12x^-7/4 Step 3: Rewrite the term with a positive exponent. (dy)/(dx) = 12x^3 - (12)/(x^7/4) (dy)/(dx) = 12x^3 - (12)/(x[4]x^3) The final answer is 12x^3 - (12)/(x[4]x^3). 3) Given y = (4x^2 + 3x + 2)/(x^3) Step 1: Divide each term in the numerator by x^3. y = (4x^2)/(x^3) + (3x)/(x^3) + (2)/(x^3) y = 4x^-1 + 3x^-2 + 2x^-3 Step 2: Differentiate each term using the power rule. (dy)/(dx) = 4(-1)x^-1-1 + 3(-2)x^-2-1 + 2(-3)x^-3-1 (dy)/(dx) = -4x^-2 - 6x^-3 - 6x^-4 Step 3: Rewrite the terms with positive exponents. (dy)/(dx) = -(4)/(x^2) - (6)/(x^3) - (6)/(x^4) The final answer is -(4)/(x^2) - (6)/(x^3) - (6)/(x^4). 4) Given y = (-2x^2)^3 Step 1: Simplify the expression. y = (-2)^3 (x^2)^3 y = -8x^2 × 3 y = -8x^6 Step 2: Differentiate using the power rule. (dy)/(dx) = -8(6)x^6-1 (dy)/(dx) = -48x^5 The final answer is -48x^5. 5) Given y = (2x^2 - 3)(3x - 2) Step 1: Expand the expression by multiplying the terms. y = (2x^2)(3x) + (2x^2)(-2) + (-3)(3x) + (-3)(-2) y = 6x^3 - 4x^2 - 9x + 6 Step 2: Differentiate each term using the power rule. (dy)/(dx) = 6(3)x^3-1 - 4(2)x^2-1 - 9(1)x^1-1 + 0 (dy)/(dx) = 18x^2 - 8x - 9x^0 (dy)/(dx) = 18x^2 - 8x - 9 The final answer is 18x^2 - 8x - 9. Send me the next one 📸