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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You're on a roll — let's differentiate these functions. h) u = ((2)/(sqrt(x)) - 1)^-1 Step 1: Rewrite the function using fractional and negative exponents. u = (2x^-1/2 - 1)^-1 Step 2: Apply the chain rule. Let w = 2x^-1/2 - 1. (du)/(dx) = -1(2x^-1/2 - 1)^-1-1 · (d)/(dx)(2x^-1/2 - 1) Step 3: Differentiate the inner function. (d)/(dx)(2x^-1/2 - 1) = 2 (-(1)/(2)x^-(1)/(2)-1) - 0 = -x^-3/2 Step 4: Substitute and simplify. (du)/(dx) = -1(2x^-1/2 - 1)^-2 (-x^-3/2) = x^-3/2(2x^-1/2 - 1)^-2 Step 5: Rewrite with positive and radical exponents. (du)/(dx) = (1)/(x^3/2)((2)/(sqrt(x)) - 1)^2 = (1)/(xsqrt(x)(2-x)sqrt(x))^2 = (1)/(xsqrt(x)(2-x))^2x = (1)/(sqrt(x)(2-x))^2 The derivative is (1)/(sqrt(x)(2-x))^2. i) v = (2x^2 - (3)/(x^2))^1/3 Step 1: Rewrite the function using negative exponents. v = (2x^2 - 3x^-2)^1/3 Step 2: Apply the chain rule. Let w = 2x^2 - 3x^-2. (dv)/(dx) = (1)/(3)(2x^2 - 3x^-2)^(1)/(3)-1 · (d)/(dx)(2x^2 - 3x^-2) Step 3: Differentiate the inner function. (d)/(dx)(2x^2 - 3x^-2) = 2(2x) - 3(-2x^-2-1) = 4x + 6x^-3 Step 4: Substitute and simplify. (dv)/(dx) = (1)/(3)(2x^2 - 3x^-2)^-2/3 (4x + 6x^-3) Step 5: Rewrite with positive exponents. (dv)/(dx) = (4x + 6)/(x^3)3(2x^2 - (3)/(x^2))^2/3 = (4x^4+6)/(x^3)3((2x^4-3)/(x^2))^2/3 = (2(2x^4+3))/(3x^3(2x^2 - 3)x^2)^2/3 The derivative is (2(2x^4+3))/(3x^3(2x^2 - 3)x^2)^2/3. j) y = sqrt(x - (1)/(x)) Step 1: Rewrite the function using fractional and negative exponents. y = (x - x^-1)^1/2 Step 2: Apply the chain rule. Let w = x - x^-1. (dy)/(dx) = (1)/(2)(x - x^-1)^(1)/(2)-1 · (d)/(dx)(x - x^-1) Step 3: Differentiate the inner function. (d)/(dx)(x - x^-1) = 1 - (-1x^-1-1) = 1 + x^-2 Step 4: Substitute and simplify. (dy)/(dx) = (1)/(2)(x - x^-1)^-1/2 (1 + x^-2) Step 5: Rewrite with positive and radical exponents. (dy)/(dx) = (1 + 1)/(x^2)2sqrt(x - (1)/(x)) = (x^2+1)/(x^2)2sqrt((x^2-1)/(x)) = (x^2+1)/(2x^2 sqrt(x^2-1))sqrt(x) = (x^2+1)/(2x^3/2)sqrt(x^2-1) The derivative is (x^2+1)/(2x^3/2)sqrt(x^2-1). k) y = (sqrt(x) - (1)/(sqrt(x)))^2 Step 1: Rewrite the function using fractional exponents. y = (x^1/2 - x^-1/2)^2 Step 2: Expand the expression first, as it simplifies nicely. y = (x^1/2)^2 - 2(x^1/2)(x^-1/2) + (x^-1/2)^2 y = x - 2x^0 + x^-1 y = x - 2 + x^-1 Step 3: Differentiate each term using the power rule. (dy)/(dx) = 1 - 0 - 1x^-1-1 = 1 - x^-2 Step 4: Rewrite with positive exponents. (dy)/(dx) = 1 - (1)/(x^2) The derivative is 1 - (1)/(x^2). l) y = 3) Step 1: Rewrite the function using fractional exponents. y = (1 - x^1/2)^1/3 Step 2: Apply the chain rule. Let w = 1 - x^1/2. (dy)/(dx) = (1)/(3)(1 - x^1/2)^(1)/(3)-1 · (d)/(dx)(1 - x^1/2) Step 3: Differentiate the inner function. (d)/(dx)(1 - x^1/2) = 0 - (1)/(2)x^(1)/(2)-1 = -(1)/(2)x^-1/2 Step 4: Substitute and simplify. (dy)/(dx) = (1)/(3)(1 - x^1/2)^-2/3 (-(1)/(2)x^-1/2) = -(1)/(6)x^-1/2(1 - x^1/2)^-2/3 Step 5: Rewrite with positive and radical exponents. (dy)/(dx) = -(1)/(6sqrt(x)(1 - x))^2/3 The derivative is -(1)/(6sqrt(x)(1 - x))^2/3. m) y = (1)/(x^2/3) - 1 Step 1: Rewrite the function using negative exponents. y = (x^2/3 - 1)^-1 Step 2: Apply the chain rule. Let w = x^2/3 - 1.