Block, SkipCaptcha, Upload, Automoderated users, Check users, delete, edit, Moderators, move, oversight, protect, rollback
19,323
edits
Changes
Jump to navigation
Jump to search
mLine 1:
Line 1: − + − + − [[category:mathematics]]+ − + + + + + + + + + + +
Differentiable function (view source)
Revision as of 21:18, September 8, 2020
, 21:18, September 8, 2020new link
A function f(x) is '''differentiable''' at the point ''a'' if and only if, as ''x'' approaches ''a'' (which it is never allowed to reach), the value of the quotient:
A [[function]] f(x) is '''differentiable''' at the point ''a'' if and only if, as ''x'' approaches ''a'' (which it is never allowed to reach), the value of the quotient:
:<math>\frac{f(x) - f(a)}{(x - a)}</math>
:<math>\frac{f(x) - f(a)}{(x - a)}</math>
approaches a limiting value that we call the [[derivative]] of the function f(x) at ''x=a''.
approaches a [[limit]]ing value that we call the [[Derivative (calculus)|derivative]] of the function f(x) at ''x=a''.
[[category:calculus]]
There is also the more rigorous <math>\epsilon-\delta</math> definition: a function f is said to be differntiable at point ''a'' if ∀<math>\epsilon>0</math> ∃<math>\delta>0</math> such that if
::<math> |x - a| < \delta\,</math>
then
::<math>|\frac{f(x) - f(a)}{x-a} - f'(a) | < \epsilon \,</math>.
[[Category:Calculus]]