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 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.

Learn how to find the derivative f'(x) for the function f(x) = -2x using first principles. Step-by-step solution includes substituting into the derivative definition, simplifying the expression, and evaluating the limit. Suitable for Grades 11-12 students learning calculus.

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