Left hand derivative and right hand derivative of a function f ( x ) at a point x = a are defined as f prime(a-)= displaystyle lim h arrow 0+ (f(a)-f(a-h)/h)= displaystyle lim h arrow 0- (f(a+h)-f(a)/h) text and f prime(a+)= displaystyle lim h arrow 0+ (f(a+h)-f(a)/h)= lim h arrow 0- (f(a)-f(a-h)/h)= displaystyle lim x arrow a+ (f(a)-f(x)/a-x) respectively Let f be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function. If f is odd, which of the following is Left hand derivative of f at x=-a

Question

Left hand derivative and right hand derivative of a function f ( x ) at a point x = a are defined as f prime(a-)= displaystyle lim h arrow 0+ (f(a)-f(a-h)/h)= displaystyle lim h arrow 0- (f(a+h)-f(a)/h) text and f prime(a+)= displaystyle lim h arrow 0+ (f(a+h)-f(a)/h)= lim h arrow 0- (f(a)-f(a-h)/h)= displaystyle lim x arrow a+ (f(a)-f(x)/a-x) respectively Let f be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function. If f is odd, which of the following is Left hand derivative of f at x=-a (A)  displaystyle lim h arrow 0- (f(a-h)-f(a)/-h) (B)  displaystyle lim h arrow 0- (f(h-a)-f(a)/h) (C)  displaystyle lim h arrow 0+ (f(a)+f(a-h)/-h) (D)  displaystyle lim h arrow 0- (f(-a)-f(-a-h)/-h)

Tardigrade Admin · Accepted Answer

L H D= displaystyle lim h arrow 0- (f(-a+h)-f(-a)/h) = displaystyle lim h arrow 0- (-f(a-h)+f(a)/h)= displaystyle lim h arrow 0- (f(a-h)-f(a)/-h)

$h \to 0^{-} lim \frac{f ( a - h ) - f ( a )}{- h}$

$h \to 0^{-} lim \frac{f ( h - a ) - f ( a )}{h}$

$h \to 0^{+} lim \frac{f ( a ) + f ( a - h )}{- h}$

$h \to 0^{-} lim \frac{f ( - a ) - f ( - a - h )}{- h}$

$L HD = h \to 0^{-} lim \frac{f ( - a + h ) - f ( - a )}{h}$
$= h \to 0^{-} lim \frac{- f ( a - h ) + f ( a )}{h} = h \to 0^{-} lim \frac{f ( a - h ) - f ( a )}{- h}$

h→0−lim​−hf(a−h)−f(a)​

h→0−lim​hf(h−a)−f(a)​

h→0+lim​−hf(a)+f(a−h)​

h→0−lim​−hf(−a)−f(−a−h)​

LHD=h→0−lim​hf(−a+h)−f(−a)​ =h→0−lim​h−f(a−h)+f(a)​=h→0−lim​−hf(a−h)−f(a)​

$h \to 0^{-} lim \frac{f ( a - h ) - f ( a )}{- h}$

$h \to 0^{-} lim \frac{f ( h - a ) - f ( a )}{h}$

$h \to 0^{+} lim \frac{f ( a ) + f ( a - h )}{- h}$

$h \to 0^{-} lim \frac{f ( - a ) - f ( - a - h )}{- h}$

$L HD = h \to 0^{-} lim \frac{f ( - a + h ) - f ( - a )}{h}$
$= h \to 0^{-} lim \frac{- f ( a - h ) + f ( a )}{h} = h \to 0^{-} lim \frac{f ( a - h ) - f ( a )}{- h}$