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Assume ( \log ) denotes the natural logarithm ( \ln ).The derivative of ( y = \ln(x^3 - x) + 7^{3x} ) is:( \frac{dy}{dx} = \frac{d}{dx}(\ln(x^3 - x)) + \frac{d}{dx}(7^{3x}) )For the first term, using the chain rule: ( \frac{d}{dx}(\ln(x^3 - x)) = \frac{1}{x^3 - x} \cdot \frac{d}{dx}(x^3 - x) = \frac{3x^2 - 1}{x^3 - x} = \frac{3x^2 - 1}{x(x^2 - 1)} )For the second term, using the rule ( \frac{d}{dx}(a^u) = a^u \ln(a) \frac{du}{dx} ): ( \frac{d}{dx}(7^{3x}) = 7^{3x} \ln(7) \cdot \frac{d}{dx}(3x) = 7^{3x} \ln(7) \cdot 3 = 3 \ln(7) 7^{3x} )So, ( \frac{dy}{dx} = \frac{3x^2 - 1}{x(x^2 - 1)} + 3 \ln(7) 7^{3x} ).Comparing this with the options, option D matches the first term and the positive sign for the second term. There appears to be a notation inconsistency with ( (2/3 \log 7) ) in the options, but assuming it represents the constant factor for ( 7^{3x} ), option D is the most consistent choice.