Use Logarithmic differentiation \( y=\left(x^{2}+1\right)^{6}(3 x-2)^{4} \)

To differentiate \( y=\left(x^{2}+1\right)^{6}(3 x-2)^{4} \) using logarithmic differentiation, first take the natural logarithm of both sides: \[ \ln(y) = \ln\left((x^{2}+1)^{6}\right) + \ln\left((3x-2)^{4}\right) \] This simplifies to: \[ \ln(y) = 6\ln(x^{2}+1) + 4\ln(3x-2) \] Now, differentiate both sides with respect to \( x \): \[ \frac{1}{y} \frac{dy}{dx} = 6 \cdot \frac{2x}{x^{2}+1} + 4 \cdot \frac{3}{3x-2} \] Now, multiply both sides by \( y \) to isolate \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = y \left( \frac{12x}{x^{2}+1} + \frac{12}{3x-2} \right) \] Finally, substitute back \( y \) to get: \[ \frac{dy}{dx} = \left(x^{2}+1\right)^{6}(3x-2)^{4} \left( \frac{12x}{x^{2}+1} + \frac{12}{3x-2} \right) \] Now you have the derivative using logarithmic differentiation!