Ucale

# Differentiability at a Point

Let f(x) be a real valued function defined on an open interval (a,b) where c (a,b). Then f(x) is said to be differentiable

or derivable at x=c or differentiable at a point

if $\displaystyle \lim _{ x\longrightarrow c }{ \frac { f\left( c-h \right) f\left( c \right) }{ \left( x-c \right) } }$

This limit is called the derivative or differential coefficient of the function f(c) at x=c, and is denoted by f'(c) or D f(c) or  $\displaystyle \frac { d }{ dx } { \left( f\left( x \right) \right) }_{ x=c }$

Thus , f(x) is differentiable at x=c

$\displaystyle \Rightarrow \lim _{ x\longrightarrow c }{ \frac { f\left( x \right) -f\left( c \right) }{ \left( x-c \right) } } \\ \Rightarrow \lim _{ x\longrightarrow { c }^{ - } }{ \frac { f\left( x \right) -f\left( c \right) }{ \left( x-c \right) } } =\lim _{ x\longrightarrow { c }^{ + } }{ \frac { f\left( x \right) -f\left( c \right) }{ \left( x-c \right) } } \\ \Rightarrow \lim _{ h\longrightarrow 0 }{ \frac { f\left( c-h \right) -f\left( c \right) }{ \left( -h \right) } } =\lim _{ h\longrightarrow 0 }{ \frac { f\left( c+h \right) -f\left( c \right) }{ \left( h \right) } } \\ Here\quad \lim _{ x\longrightarrow { c }^{ - } }{ \frac { f\left( x \right) -f\left( c \right) }{ \left( x-c \right) } } =\lim _{ h\longrightarrow { 0 } }{ \frac { f\left( x \right) -f\left( c \right) }{ \left( x-c \right) } }$

is called the left hand derivative of f(x) at x=c and is denoted by f'(c-) or LF'(c)

$\displaystyle while\lim _{ x\longrightarrow { c }^{ - } }{ \frac { f\left( x \right) -f\left( c \right) }{ \left( x-c \right) } } =\lim _{ x\longrightarrow { c }^{ + } }{ \frac { f\left( x \right) -f\left( c \right) }{ \left( x-c \right) } }$

is called the right hand derivative of f(x) at x=c and is called the right hand derivative of f(x) at x=c and is denoted by f'(c+) and Rf'(c).

Thus, f(x) is differentiable at x=c

Lf'(c) = Rf'(c)

$\displaystyle IfLf'(c)\neq Rf'(c)$ , we say that f(x) is not differentiable at x=c.

April 18, 2019
Which class you are presently in?
Choose an option. You can change your option at any time.
You will be solving questions and growing your critical thinking skills.