To determine the derivative of the function $f(x) = -2x$ from first principles, we use the definition of the derivative:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.$

Step 1: Find $f(x+h)$

Given $f(x) = -2x$, we substitute $x+h$ into the function:

$f(x+h) = -2(x+h).$

Step 2: Calculate $f(x+h) - f(x)$

Now, compute $f(x+h) - f(x)$:

$f(x+h) - f(x) = [-2(x+h)] - (-2x).$

Simplify the expression:

$f(x+h) - f(x) = (-2x - 2h) + 2x.$

The $-2x$ and $+2x$ cancel each other out:

$f(x+h) - f(x) = -2h.$

So, the expression for $f(x+h) - f(x)$ is:

$f(x+h) - f(x) = -2h.$

Step 3: Compute the Derivative Using the Limit Definition

Substitute the expression we found into the limit definition of the derivative:

$f'(x) = \lim_{h \to 0} \frac{-2h}{h}.$

Simplify the fraction:

$f'(x) = \lim_{h \to 0} (-2) = -2.$

Conclusion

The derivative of $f(x) = -2x$ is $f'(x) = -2$.

Based on our working, the correct expression for $f(x+h) - f(x)$ is:

$f(x+h) - f(x) = -2h.$

Would you like more details on any step?

Here are 5 related questions you might consider:

1. How do you apply the first principle to find the derivative of more complex functions?
2. Why does the expression for $f'(x)$ simplify to just a constant in this case?
3. How can we interpret the derivative $f'(x) = -2$ geometrically?
4. What happens to the derivative if the function is $f(x) = -2x^2$?
5. How does this method of finding derivatives compare with using differentiation rules?

Tip: The derivative represents the rate of change of a function; understanding its geometric interpretation can enhance comprehension.