# What is the derivative of f(x)= log_3(3-2x)?

Jan 9, 2016

$f ' \left(x\right) = \frac{2}{\left(2 x - 3\right) \ln 3}$

#### Explanation:

Use the change of base formula to express $f \left(x\right)$ in terms of the natural logarithm.

$f \left(x\right) = \ln \frac{3 - 2 x}{\ln} 3$

Thus,

$f ' \left(x\right) = \frac{1}{\ln} 3 \cdot \frac{d}{\mathrm{dx}} \left(\ln \left(3 - 2 x\right)\right)$

To find the derivative of a natural logarithm function, use the chain rule:

$\frac{d}{\mathrm{dx}} \left(\ln \left(u\right)\right) = \frac{1}{u} \cdot u '$

Thus,

$f ' \left(x\right) = \frac{1}{\ln} 3 \cdot \frac{1}{3 - 2 x} \cdot \frac{d}{\mathrm{dx}} \left(3 - 2 x\right)$

$= \frac{1}{\ln} 3 \cdot \frac{1}{3 - 2 x} \cdot - 2$

$= \frac{- 2}{\left(3 - 2 x\right) \ln 3}$

$= \frac{2}{\left(2 x - 3\right) \ln 3}$