Question Detail

We need to differentiate each term separately.First term: y1 = log(x3 - x). Assuming log is the natural logarithm (ln):Recall d/dx [ln(u)] = (1/u) * u'.Let u = x3 - x, then u' = 3x2 - 1.dy1/dx = (1/(x3 - x)) * (3x2 - 1)Factor the denominator: x3 - x = x(x2 - 1).dy1/dx = (3x2 - 1) / (x(x2 - 1))Second term: y2 = 73/4 x.Recall d/dx [au] = au * ln(a) * u'.Let a = 7 and u = (3/4)x, then u' = 3/4.dy2/dx = 73/4 x * ln(7) * (3/4)However, the options provided use 'log 7' and '2/3' instead of 'ln 7' and '3/4'. This suggests a potential typo in either the question or options regarding the base of the logarithm and the coefficient of x in the exponent. Assuming the options are the intended derivatives: if the exponent was (2/3)x and 'log' implies ln, then the derivative of the second term would be 72/3 x * ln(7) * (2/3).If we assume the intent was to match the provided option D's structure, then we must assume the question means:dy/dx = (3x2 - 1) / (x(x2 - 1)) + 73/4 x * (2/3 log 7)This is problematic as the derivative of 73/4 x is 73/4 x * ln(7) * (3/4). There is a clear mismatch in the constant coefficient (3/4 vs 2/3) and the base of the log (ln 7 vs log 7). However, among the given options, D is the most likely intended answer if we overlook the numerical constant mismatch (3/4 vs 2/3) and assume 'log 7' is a symbolic representation of a constant related to ln(7).Given the direct mathematical derivation matches (3x2-1)/(x(x2-1)) for the first part and the general form for the second part (7exponent * constant), Option D is structurally the closest if we accept the (2/3 log 7) as the intended derivative constant for the exponential term despite the initial exponent given.Let's assume the question implicitly meant the derivative constant for the exponential part is (2/3 log 7), then D is the answer.