Use Logarithmic differentiation \( y=\frac{x^{\frac{1}{4}} \sqrt{x^{3}+5}}{\left(3 x^{2}-1\right)^{4}} \)

To apply logarithmic differentiation, we first take the natural logarithm of both sides: \[ \ln(y) = \ln\left(\frac{x^{\frac{1}{4}} \sqrt{x^{3}+5}}{(3 x^{2}-1)^{4}}\right) \] Using the properties of logarithms, we can simplify this to: \[ \ln(y) = \ln(x^{\frac{1}{4}}) + \ln(\sqrt{x^{3}+5}) - 4\ln(3x^{2}-1) \] Now, we differentiate both sides with respect to \( x \): \[ \frac{1}{y} \frac{dy}{dx} = \frac{1}{4x} + \frac{1}{2(x^{3}+5)}(3x^{2}) - 4\left(\frac{6x}{3x^{2}-1}\right) \] Now to isolate \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = y \left( \frac{1}{4x} + \frac{3x^{2}}{2(x^{3}+5)} - \frac{24x}{3x^{2}-1} \right) \] Substituting back the original \( y \): \[ \frac{dy}{dx} = \frac{x^{\frac{1}{4}} \sqrt{x^{3}+5}}{(3 x^{2}-1)^{4}} \left( \frac{1}{4x} + \frac{3x^{2}}{2(x^{3}+5)} - \frac{24x}{3x^{2}-1} \right) \] And that gives you the derivative \( \frac{dy}{dx} \) using logarithmic differentiation!