How do you differentiate f(x)=sqrt(1/(3x-2)) using the chain rule?

Dec 5, 2015

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

Explanation:

Rewire the expression using the rational exponential rule

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

Rewrite using power rule of exponent
$f \left(x\right) = {\left(3 x - 2\right)}^{- \frac{1}{2}}$

Begin to find the derivative
$f ' \left(x\right) = - \frac{1}{2} {\left(3 x - 2\right)}^{- \frac{1}{2} - 1} \cdot 3$

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

Rewrite without negative exponent
$f ' \left(x\right) = - \frac{1}{2 {\left(3 x - 2\right)}^{\frac{3}{2}}}$