If f (x) is odd and differentiable , then f'(x) is

Answer

To determine whether \( f'(x) \) is even or odd given that \( f(x) \) is an odd and differentiable function, we can follow these steps:

### Step 1: Understand the definition of an odd function
An odd function satisfies the property:
\[ f(-x) = -f(x) \]
for all \( x \).

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

### Step 3: Find the derivative at \( -x \)
We need to find \( f'(-x) \):
\[ f'(-x) = \lim_{h \to 0} \frac{f(-x+h) - f(-x)}{h} \]

### Step 4: Substitute the property of odd functions
Using the property of odd functions, we have:
\[ f(-x+h) = -f(x-h) \quad \text{and} \quad f(-x) = -f(x) \]
Thus, we can rewrite \( f'(-x) \):
\[ f'(-x) = \lim_{h \to 0} \frac{-f(x-h) + f(x)}{h} \]

### Step 5: Factor out the negative sign
This can be rearranged as:
\[ f'(-x) = -\lim_{h \to 0} \frac{f(x-h) - f(x)}{h} \]

### Step 6: Change the variable in the limit
Let \( k = -h \), then as \( h \to 0 \), \( k \to 0 \):
\[ f'(-x) = -\lim_{k \to 0} \frac{f(x+k) - f(x)}{-k} = -f'(x) \]

### Step 7: Conclusion
Since we have shown that:
\[ f'(-x) = -f'(x) \]
This means that \( f'(x) \) is an odd function.

### Final Answer
Thus, \( f'(x) \) is odd.

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