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Question
Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined asf′(a−)=limh→0+f(a)−f(a−h)h=limh→0−f(a)−f(a−h)h=limx→a+f(a)−f(x)a−x respectivelyLet 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 even, which of the following is Right hand derivative of f′ at x=a
Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined as
f′(a−)=limh→0+f(a)−f(a−h)h=limh→0−f(a)−f(a−h)h=limx→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.
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Solution
The correct option is A limh→0−f′(a)+f′(−a+h)h
the right hand derivative of f′ at x=a will be given by
limh→0+f′(a)−f′(a−h)h
In the above can f is even, hence f′(x) will be odd.
Thus
f′(−x)=−f′(x)
Thus
f′(a−h)=f′(−(h−a))
=−f′(h−a)
Substituting in the above equation gives us
limh→0+f′(a)−f′(a−h)h
=limh→0+f′(a)+f′(h−a)h
=limh→0+f′(a)+f′(−a+h)h
the right hand derivative of f′ at x=a will be given by
limh→0+f′(a)−f′(a−h)h
In the above can f is even, hence f′(x) will be odd.
Thus
f′(−x)=−f′(x)
Thus
f′(a−h)=f′(−(h−a))
=−f′(h−a)
Substituting in the above equation gives us
limh→0+f′(a)−f′(a−h)h
=limh→0+f′(a)+f′(h−a)h
=limh→0+f′(a)+f′(−a+h)h
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