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

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

• A. 2x/ln(x)
• B. (2x - x)/ln^2(x)
• C. 2x/ln(x) - x/ln^2(x)
• D. (x^2 + 1)/ln^2(x)

Answer: (C)

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}

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}