If y = x^2/ln(x), what is the expression for dy/dx?

To find the derivative of the function ( y = \frac{x^2}{\ln(x)} ), we can apply the quotient rule, which states that if you have a function ( y = \frac{u}{v} ), the derivative is given by:

[

\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}

]

In this context, ( u = x^2 ) and ( v = \ln(x) ).

First, we differentiate ( u ) and ( v):

1. The derivative of ( u = x^2 ) is ( \frac{du}{dx} = 2x ).

2. The derivative of ( v = \ln(x) ) is ( \frac{dv}{dx} = \frac{1}{x} ).

Substituting these into the quotient rule formula gives us:

[

\frac{dy}{dx} = \frac{\ln(x) \cdot (2x) - x^2 \cdot \left(\frac{1}{x}\right)}{(\ln(x))^2}