Question Detail

First, differentiate y = log(x3 - x). Assuming 'log' refers to the natural logarithm (ln), the derivative is (1⁄(x3-x)) * (3x2-1) using the chain rule. This simplifies to 3x2-1⁄x(x2-1). Next, differentiate 7(3⁄2)x. The derivative of au (where u is a function of x) is au * ln a * du⁄dx. Here, a=7 and u=(3⁄2)x, so du⁄dx = 3⁄2. Thus, the derivative is 7(3⁄2)x * ln 7 * (3⁄2). Combining these, dy⁄dx = 3x2-1⁄x(x2-1) + (3⁄2) * 7(3⁄2)x * ln 7. Option 'D' provides the correct first term. The second term, '7(3⁄2)x (2⁄3 log 7)', appears to have a typo in the coefficient (should be 3⁄2 not 2⁄3) and potentially the base of the logarithm (should be ln 7 if it's natural log, or ln(7)⁄ln(10) if 'log' means base 10). Assuming the intended term matches the calculated derivative, option D is the best fit.