Thread: diff. derivative

Def: Let f be a real-valued function defined on an interval I containing the point c, (we allow the possibilty that c is an endpoint of I) we say that f is differentiable at c (or has a derivative at c) if the limit

lim x->c (f(x) - f(c))/(x-c)
exists and is finite.

use the definition above to find the derivative of each function.

C) f(x) = 1/x for x is not equal to 0
D) f(x) = sqrt(x) for x>0

Originally Posted by learn18

Def: Let f be a real-valued function defined on an interval I containing the point c, (we allow the possibilty that c is an endpoint of I) we say that f is differentiable at c (or has a derivative at c) if the limit

You should say I is an open interval. And c cannot be an endpoint. Otherwise the limit might not make sense.

C) f(x) = 1/x for x is not equal to 0

Let y!=0 then what is derivative at y, if it exists.

(f(x) - f(y))/(x-y) = (1/x - 1/y)/(x-y) = (-(x-y)/xy)/(x-y) = -1/xy
As x--> y we have,
-1/y^2
Thus,
-1/x^2 is the derivative.