Question Detail

Assuming the function intended was $y = \ln(x^3 - x) + 7^{\frac{2}{3}x}$ and $\log 7$ in the options means $\ln 7$:First term: Differentiate $f(x) = \ln(x^3 - x)$ using the chain rule $\frac{d}{dx}(\ln u) = \frac{1}{u}\frac{du}{dx}$:$\frac{d}{dx}(\ln(x^3 - x)) = \frac{1}{x^3 - x} \cdot \frac{d}{dx}(x^3 - x) = \frac{3x^2 - 1}{x(x^2 - 1)}$Second term: Differentiate $g(x) = 7^{\frac{2}{3}x}$ using the rule $\frac{d}{dx}(a^u) = a^u \cdot \ln a \cdot \frac{du}{dx}$:$\frac{d}{dx}(7^{\frac{2}{3}x}) = 7^{\frac{2}{3}x} \cdot \ln 7 \cdot \frac{d}{dx}(\frac{2}{3}x) = 7^{\frac{2}{3}x} \cdot \ln 7 \cdot \frac{2}{3}$Combining both derivatives:$\frac{dy}{dx} = \frac{3x^2 - 1}{x(x^2 - 1)} + 7^{\frac{2}{3}x} \cdot \frac{2}{3} \ln 7$This matches option D, assuming $7^{3x}$ in the question is a typo for $7^{\frac{2}{3}x}$.