analysis I

• Nov 2nd 2006, 10:32 AM
gap135
analysis I
suppose that f is differentiable at a and f(a) is not equal to 0.
show that for h sufficiently small f(a+h) is not equal to zero and using the definition of a derivative directly, prove that 1/(f(x)) is differentiable at x=a and
(1/f)'(a) = -f'(a)/f^2(a)

stuck on this one anyone have any ideas? is the second part just plug and chug?
• Nov 2nd 2006, 10:50 AM
Plato
Quote:

Originally Posted by gap135
suppose that f is differentiable at a and f(a) is not equal to 0.

Do you mean that \$\displaystyle f'(a) \not= 0\$?
Otherwise, because f is differentiable at a then f is continuous at a.
If f is not zero at a then there is a neighborhood of a throughout which f is not zero.
• Nov 2nd 2006, 11:28 AM
gap135
re
It's f(a) not f'(a)
• Nov 2nd 2006, 11:56 AM
Plato
Then the first part is trival. If f is contiouous at a and f(a) is not zero then if (x) is not zero on some neighborhood of a.
• Nov 2nd 2006, 05:05 PM
ThePerfectHacker
Quote:

Originally Posted by gap135
, prove that 1/(f(x)) is differentiable at x=a and
(1/f)'(a) = -f'(a)/f^2(a)
?

What does it means (1/f)'(a)?