Question Detail

Differentiate each term separately.For the first term, y1 = log(x3 - x). Assuming 'log' is the natural logarithm (ln):d/dx (ln(x3 - x)) = (d/dx(x3 - x))⁄(x3 - x) = (3x2 - 1)⁄(x(x2 - 1)).For the second term, y2 = 73/x = 73x-1.Use the rule d/dx(af(x)) = af(x) ln(a) f'(x).Here, a=7, f(x) = 3x-1, and f'(x) = -3x-2.d/dx (73x-1) = 73x-1 ln(7) (-3x-2) = -(3ln(7))⁄x2 73/x.So, dy/dx = (3x2 - 1)⁄(x(x2 - 1)) - (3ln(7))⁄x2 73/x.Comparing with the options, the first part matches (3x2-1)⁄x(x2-1) for options A and D. The second part must be negative, which rules out D. Option A has a negative sign for the second term, although the constant part (2⁄3 log7) differs from the calculated -(3ln(7))⁄x2. Given the choices, option A is the best match for the derived signs and structure, assuming the constant (2/3 log7) is a specific simplification or representation of ln(7) and 1/x2 part for this problem, or there might be a minor typo in the numerical factor.