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

f′(x)   if   f(x)=−2x.
Then based on your working, what is the correct expression for

 
f(x+h)−f(x)?

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.

Learn how to compute the derivative of the function f(x) = -2x using first principles. This detailed solution covers the definition of derivatives, step-by-step calculations for f(x+h) - f(x), and the interpretation of results. Suitable for advanced high school students studying calculus.

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