The Most Common Mistake Students Make with the Limit Definition of the Derivative

If you've ever tried using the limit definition of a derivative and ended up with a totally wrong answer — you're not alone.

In fact, there's one mistake that shows up again and again, even with students who are good at algebra:

They forget to distribute the negative sign when subtracting $f(x)$.

Let's walk through a classic example and break it down.

📌 The Limit Definition of the Derivative

The definition is:

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

Looks simple, right? But it's a landmine if you're not careful with your signs.

We'll use the function:

$$f(x) = 3 - x^2$$

Let's compute the derivative using the limit definition — and watch where things go wrong.

❌ The Incorrect Way (What Many Students Do)

Step 1: Plug into the definition

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

Now compute $f(x+h)$ and $f(x)$:

$$f(x+h) = 3 - (x+h)^2$$ $$f(x) = 3 - x^2$$

So we plug in:

$$f'(x) = \lim_{h \to 0} \frac{[3 - (x+h)^2] - (3 - x^2)}{h}$$

Step 2: The Mistake Happens Here

Students often write:

$$\color{red}{= \lim_{h \to 0} \frac{3 - (x+h)^2 - 3 - x^2}{h}}$$

🔴 Wrong! They dropped the parentheses around $f(x)$, so the minus didn't apply to both terms in $(3 - x^2)$.

That simplifies to:

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

...which gives the wrong derivative.

✅ The Correct Way

Let's go back to:

$$f'(x) = \lim_{h \to 0} \frac{[3 - (x+h)^2] - (3 - x^2)}{h}$$

Now distribute the negative properly across the second set of parentheses:

$$= \lim_{h \to 0} \frac{3 - (x+h)^2 - 3 + x^2}{h}$$

Now simplify:

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

Now expand $(x+h)^2$:

$$= \lim_{h \to 0} \frac{-[x^2 + 2xh + h^2] + x^2}{h}$$ $$= \lim_{h \to 0} \frac{-x^2 - 2xh - h^2 + x^2}{h}$$

Now combine like terms:

$$= \lim_{h \to 0} \frac{-2xh - h^2}{h}$$

Factor out $h$:

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

Cancel $h$:

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

Now take the limit:

$$= -2x$$

✅ That's the correct derivative of $f(x) = 3 - x^2$.

📊 Side-by-Side Comparison

Step	Incorrect (Common Mistake)	Correct
Subtraction Step	$3 - (x+h)^2 - 3 - x^2$ ⛔ minus not distributed	$3 - (x+h)^2 - (3 - x^2)$ ✅ minus distributed
Simplified	$-(x+h)^2 + x^2$ gets wrong answer	Leads to $-2x$

🧠 Final Tip

If you're subtracting something like $f(x) = 3 - x^2$, always wrap it in parentheses when plugging into the formula.

The subtraction is of the entire expression — not just the first term!