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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
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