Find, from first principles, the derivative of the following w.r.t. x :
-x

Answer

To find the derivative of the function \( f(x) = -x \) from first principles, we will use 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 1: Identify the function
We have the function:
\[ f(x) = -x \]

### Step 2: Calculate \( f(x+h) \)
Now, we need to find \( f(x+h) \):
\[ f(x+h) = -(x+h) = -x - h \]

### Step 3: Substitute into the derivative formula
Now, we substitute \( f(x+h) \) and \( f(x) \) into the derivative formula:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{(-x - h) - (-x)}{h} \]

### Step 4: Simplify the expression
Simplifying the expression inside the limit:
\[ f'(x) = \lim_{h \to 0} \frac{-x - h + x}{h} = \lim_{h \to 0} \frac{-h}{h} \]

### Step 5: Further simplify
This simplifies to:
\[ f'(x) = \lim_{h \to 0} -1 \]

### Step 6: Evaluate the limit
Since the limit does not depend on \( h \), we can directly evaluate it:
\[ f'(x) = -1 \]

### Conclusion
Thus, the derivative of \( f(x) = -x \) with respect to \( x \) is:
\[ f'(x) = -1 \]