if the right hand derivative of h(x) = {x} ( {.} is fractional of x) exists at x =1 and it is equal to _____\

Answer

To find the right-hand derivative of the function \( h(x) = \{x\} \) (where \(\{x\}\) denotes the fractional part of \(x\)) at \(x = 1\), we can follow these steps:

### Step 1: Understand the Right-Hand Derivative
The right-hand derivative of a function \( h(x) \) at a point \( a \) is defined as:
\[ h'(a^+) = \lim_{h \to 0^+} \frac{h(a + h) - h(a)}{h} \]
In our case, we need to find \( h'(1^+) \).

### Step 2: Substitute into the Derivative Formula
We substitute \( a = 1 \):
\[ h'(1^+) = \lim_{h \to 0^+} \frac{h(1 + h) - h(1)}{h} \]

### Step 3: Calculate \( h(1) \) and \( h(1 + h) \)
We know that:
- \( h(1) = \{1\} = 0 \) (since the fractional part of 1 is 0)
- For \( h(1 + h) \), we have:
\[ h(1 + h) = \{1 + h\} = 1 + h - 1 = h \quad \text{(for small } h \text{ where } 0 < h < 1\text{)} \]

### Step 4: Substitute Values into the Derivative Formula
Now, we can substitute these values into our derivative formula:
\[ h'(1^+) = \lim_{h \to 0^+} \frac{h - 0}{h} = \lim_{h \to 0^+} \frac{h}{h} \]

### Step 5: Simplify the Expression
The expression simplifies to:
\[ h'(1^+) = \lim_{h \to 0^+} 1 = 1 \]

### Conclusion
Thus, the right-hand derivative of \( h(x) \) at \( x = 1 \) exists and is equal to:
\[ \boxed{1} \]

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