Left hand derivative and right hand derivative of a function `f(x)` at a point `x=a` are defined as
`f'(a^-)=lim_(hrarr0^(+))(f(a)-f(a-h))/(h)`
`=lim_(hrarr0^(+))(f(a+h)-f(a))/(h)`
`andf'(a^(+))=lim_(hrarr0^(+))(f(a+h)-f(a))/(h)`
`=lim_(hrarr0^(+))(f(a)-f(a+h))/(h)`
`=lim_(hrarr0^(+)) (f(a)-f(x))/(a-x)` respectively.
Let `f` be a twice differentiable function. We also know that derivative of a even function is odd function and derivative of an odd function is even function.
If `f` is even function, which of the following is right hand derivative of `f'` at `x=a?`

Options

A. (a) `underset(hrarr0^(-))lim(f'(a)+f'(-a+h))/(-h)`
B. (b) `underset(hrarr0^(+))lim(f'(a)+f'(-a-h))/(h)`
C. (c) `underset(hrarr0^(-))lim(-f'(-a)+f'(-a-h))/(-h)`
D. (d) `underset(hrarr0^(+))lim(f'(a)+f'(-a+h))/(-h)`

Answer

To find the right-hand derivative of the function \( f' \) at \( x = a \) given that \( f \) is an even function, we can follow these steps:

### Step 1: Understand the properties of even and odd functions
Since \( f \) is an even function, we know that:
\[ f(-x) = f(x) \quad \text{for all } x. \]
The derivative of an even function, \( f' \), is an odd function. This means:
\[ f'(-x) = -f'(x) \quad \text{for all } x. \]

### Step 2: Write the definition of the right-hand derivative
The right-hand derivative of \( f' \) at \( x = a \) is defined as:
\[ f''(a^+) = \lim_{h \to 0^+} \frac{f'(a + h) - f'(a)}{h}. \]

### Step 3: Substitute the property of the odd function
Using the property of the odd function, we can express \( f'(a + h) \) in terms of \( f'(-a - h) \):
\[ f'(a + h) = -f'(-a - h). \]
Now, substitute this into the right-hand derivative definition:
\[ f''(a^+) = \lim_{h \to 0^+} \frac{-f'(-a - h) - f'(a)}{h}. \]

### Step 4: Simplify the expression
We can rewrite the limit:
\[ f''(a^+) = \lim_{h \to 0^+} \frac{-f'(-a - h) + (-f'(-a))}{h}. \]
This simplifies to:
\[ f''(a^+) = \lim_{h \to 0^+} \frac{-f'(-a - h) + f'(-a)}{h}. \]

### Step 5: Recognize the limit as a derivative
The expression we have is now in the form of a derivative:
\[ f''(a^+) = \lim_{h \to 0^+} \frac{f'(-a) - f'(-a - h)}{h}. \]
This is the definition of the derivative of \( f' \) at \( -a \), which can be expressed as:
\[ f''(a^+) = -f''(-a). \]

### Conclusion
Thus, the right-hand derivative of \( f' \) at \( x = a \) can be expressed as:
\[ f''(a^+) = \lim_{h \to 0^+} \frac{f'(a) - f'(-a - h)}{h}. \]
This matches with option B:
\[ \text{B. } \lim_{h \to 0^+} \frac{f'(a) - f'(-a - h)}{h}. \]