On a piece of paper, determine from the first principles,

f′(x)   if   f(x)=−2x.
Then based on your working, answer the following question. One of the steps you need to do before evaluating the limit as h approaches 0 is to cancel h in the numerator and denominator. When this is done, the correct expression obtained is ----

To determine the derivative of \( f(x) = -2x \) from first principles, we start by using the definition of the derivative:

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

Given \( f(x) = -2x \), we substitute into the definition:

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

Simplify the expression inside the limit:

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

Notice that the \( -2x \) and \( 2x \) terms cancel each other out:

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

Now, cancel the \( h \) in the numerator and the denominator:

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

Since \( -2 \) is a constant and independent of \( h \), the limit is simply:

\[
f'(x) = -2
\]

### Answer to the Given Question:
The correct expression obtained after canceling \( h \) in the numerator and denominator is \( -2 \).

Would you like to go into more detail about any step? Here are five related questions to expand on this topic:

1. How does the derivative from first principles apply to more complex functions?
2. Can you explain the geometric interpretation of the derivative?
3. How does the derivative of \( f(x) = -2x \) compare to the derivative of other linear functions?
4. What happens if we have a higher-degree polynomial? How does first principles differ?
5. How do you find the derivative of a function that is not linear using first principles?

**Tip:** When simplifying expressions during differentiation from first principles, always ensure that any terms involving \( h \) can be canceled before taking the limit as \( h \) approaches zero.

Learn how to find the derivative f'(x) of the function f(x) = -2x using first principles. Step-by-step solution provided, including canceling h in the numerator and denominator. Suitable for high school students studying calculus.

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