Question

f(x)= 1/(x^2), x>0 if g(x) is the inverse of f(x), find g'(2)
a. -4 sqrt2
b. -1/ (sqrt2)
c. -sqrt2
d. -(sqrt2)/8
e. -8
PLEASE HELP AND SHOW ALL STEPS! THANKS :)

Answer 1

if g(x) is the inverse of f(x), then
\[g(x) = \sqrt{\frac{1}{x}} \]
so
\[g'(x) = -\frac{1}{2}x^{-\frac{3}{2}} \]
and
\[g'(2) = -\frac{1}{2}(2)^{-\frac{3}{2}} = -2^{-\frac{5}{2}} = - (\frac{1}{4})(2^{-\frac{1}{2}})=-\frac{1}{4\sqrt{2}}=-\frac{\sqrt{2}}{8}\]