# How do you find derivative of y=(-2lnx)/(3x-1)?

Mar 6, 2015

How: Use the quotient rule and the fact that $\frac{d}{\mathrm{dx}} \left(\ln x\right) = \frac{1}{x}$.

The quotient rule for derivatives says that: $\frac{d}{\mathrm{dx}} \left(\frac{N}{D}\right) = \frac{N ' D - N D '}{D} ^ 2$

So, $y = \frac{- 2 \ln x}{3 x - 1}$ has derivative:

$y ' = \frac{- 2 \left(\frac{1}{x}\right) \left(3 x - 1\right) - \left(- 2 \ln x\right) \left(3\right)}{3 x - 1} ^ 2$ (we're finished with the calculus).

Use algebra to re-write:

$y ' = \frac{\left(\frac{- 2 \left(3 x - 1\right)}{x}\right) + 6 \ln x}{3 x - 1} ^ 2 = \frac{\frac{2 - 6 x + 6 x \ln x}{x}}{3 x - 1} ^ 2$

$y ' = \frac{2 - 6 x + 6 x \ln x}{x {\left(3 x - 1\right)}^{2}}$