Question Detail

Let's assume 'log' denotes the natural logarithm 'ln'. The problem statement for 7 to the power of 3/x and the (2/3 log7) factor in the options suggest a possible typo, where 7(3/x) might have been intended as 7(2x/3) for the given options to be mathematically consistent. Assuming 'log' is 'ln' and the factor (2/3 log7) for the second term is correct (implying the derivative of 7^(2x/3)), we proceed as follows:1. Differentiate the first term, y1 = ln(x3 - x): dy1/dx = (1 / (x3 - x)) * d/dx(x3 - x) = (3x2 - 1) / (x(x2 - 1))2. Differentiate the second term, y2 = 7(3/x). If we assume it should be 7(2x/3) to match the options' derivative form: dy2/dx = 7(2x/3) * ln(7) * d/dx(2x/3) = 7(2x/3) * ln(7) * (2/3) Given the options use 7(3/x) and a factor of (2/3 log7), we choose the option that has the correct first term and the given factor for the second term, acknowledging the likely typo in the exponent if the constant (2/3 log7) is correctly derived.Combining the terms, we get dy/dx = (3x2 - 1) / (x(x2 - 1)) + 7(3/x) (2/3 log7).