# If f(x) =-sqrt(3x-1)  and g(x) = (2-1/x)^2 , what is f'(g(x)) ?

Sep 15, 2017

$- \frac{3 {x}^{2}}{22 {x}^{2} - 12 x + 6}$

#### Explanation:

First of all, figure out what $f ' \left(x\right)$ is.
$\frac{d}{\mathrm{dx}} \left(- \sqrt{3 x - 1}\right)$ = $- \frac{d}{\mathrm{dx}} \left(\sqrt{3 x - 1}\right)$.
Using the chain rule, we get:
$- \frac{1}{2 \left(3 x - 1\right)} \cdot 3$ = $- \frac{3}{6 x - 2}$.
Now, we just plug $g \left(x\right)$ in, and we get the answer.
$- \frac{3}{6 {\left(2 - \frac{1}{x}\right)}^{2} - 2}$
Expand the above equation...
$- \frac{3}{6 \left(4 - \frac{2}{x} + \frac{1}{x} ^ 2\right) - 2}$
Expand and simplify.
$- \frac{3}{24 - \frac{12}{x} + \frac{6}{x} ^ 2 - 2}$ = $- \frac{3}{22 - \frac{12}{x} + \frac{6}{x} ^ 2}$
Move all the fractions in the denominator into one fraction.
$- \frac{3}{\frac{22 {x}^{2} - 12 x + 6}{x} ^ 2}$
Move the denominator of this new fraction into the numerator of the bigger fraction.
$- \frac{3 {x}^{2}}{22 {x}^{2} - 12 x + 6}$
This bigger fraction cannot be simplified further.