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: Substitute $f(x) = -2x$ into the definition of the derivative.

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

Step 2: Simplify the expression inside the limit.

Expand and simplify the expression in the numerator:

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

The $-2x$ and $+2x$ terms cancel out:

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

Step 3: Cancel $h$ in the numerator and denominator.

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

Step 4: Evaluate the limit as $h$ approaches 0.

Since the expression $-2$ does not depend on $h$, the limit is simply:

$f'(x) = -2$

Final Answer:

The correct expression obtained after canceling $h$ in the numerator and denominator is $-2$.

Would you like more details or have any questions? Here are some related problems to consider:

1. Find the derivative of $f(x) = 3x^2$ using first principles.
2. Determine the derivative of $f(x) = \sqrt{x}$ from first principles.
3. Use the first principles to differentiate $f(x) = \frac{1}{x}$.
4. Calculate the derivative of $f(x) = x^3$ using the definition of the derivative.
5. Differentiate $f(x) = \sin(x)$ from first principles.

Tip: When using the first principles method to find a derivative, simplifying the numerator as much as possible before canceling terms helps avoid mistakes in the final steps.