Chapter Three The Derivatives
Chapter Three The Derivatives
Definition of The Derivative Function The function f defined by the formula f(x)=lim I(x+h)-f(x) h-→0 h which is called the derivative of fwith respect to x. The domain of f consists of all x in the domain of ffor which the limit exists
Definition of The Derivative Function ⚫ The function f’ defined by the formula which is called the derivative of f with respect to x. The domain of f’ consists of all x in the domain of f for which the limit exists. 0 ( ) ( ) '( ) limh f x h f x f x → h + − =
Definition of Differentiability A function fis said to be differentiable at xo if the limit f()=limh-f(x) h-→0 h exists.If fis differentiable at each point of the open interval of the form (a,b),then we say that it is differentiable on (a,b),and similarly for open intervals of the form (a,+oo)and (o,+o). The last case,we say that fis differentiable everywhere
Definition of Differentiability ⚫ A function f is said to be differentiable at x0 if the limit exists. If f is differentiable at each point of the open interval of the form (a,b), then we say that it is differentiable on (a,b), and similarly for open intervals of the form (a,+∞) and (-∞,+∞). The last case, we say that f is differentiable everywhere. 0 0 0 0 ( ) ( ) '( ) limh f x h f x f x → h + − =
Differentiable Point VS Continuous Point ●Theorem If a function fis differentiable at xo,then fis continuous at xo
Differentiable Point VS Continuous Point ⚫Theorem If a function f is differentiable at x0 , then f is continuous at x0
One-sided Derivative Left-hand derivatives L()=lim-f(xo) h>0 h Right-hand derivatives f.()=lim f(h)-f(xo) h>0 h
One-sided Derivative ⚫ Left-hand derivatives ⚫ Right-hand derivatives 0 0 0 0 ( ) ( ) '( ) lim h f x h f x f x h − → − + − = 0 0 0 0 ( ) ( ) '( ) lim h f x h f x f x h + → + + − =