# How do you differentiate f(x)=sqrt(((3x)/(2x-3)) using the chain rule?

Jun 1, 2018

$f ' \left(x\right) = - \frac{9}{2} \cdot {\left(\frac{3 \cdot x}{2 \cdot x - 3}\right)}^{- \frac{1}{2}}$

#### Explanation:

Writing

$f \left(x\right) = {\left(\frac{3 \cdot x}{2 x - 3}\right)}^{\frac{1}{2}}$

so we get

$f ' \left(x\right) = \frac{1}{2} \cdot {\left(\frac{3 \cdot x}{2 x - 3}\right)}^{- \frac{1}{2}} \cdot \frac{3 \cdot \left(2 x - 3\right) - 3 x \cdot 2}{2 x - 3} ^ 2$

so

$f ' \left(x\right) = - \frac{9}{2} \cdot {\left(\frac{3 x}{2 x - 3}\right)}^{- \frac{1}{2}}$