Find dy/dx of x³ using the limit definition of derivative.

Derivative of x³ from First Principles

Question

Find the derivative of $x^3$ from first principles (i.e., using the limit definition of the derivative).

This appeared in CBSE 2024 Board Exam and is a standard 3-mark question. The first principles method shows up every year — know it cold.

Solution — Step by Step

The derivative from first principles is:

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

Here, $f(x) = x^3$ , so we substitute $f(x+h) = (x+h)^3$ .

Use the identity $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ :

(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3

Don’t skip this expansion — it’s where most marks are awarded in board exams.

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

= 3x^2h + 3xh^2 + h^3

The $x^3$ terms cancel cleanly — this always happens and is a good self-check.

\frac{f(x+h) - f(x)}{h} = \frac{3x^2h + 3xh^2 + h^3}{h}

Factor out $h$ from the numerator and cancel:

= 3x^2 + 3xh + h^2

f'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2)

Every term with $h$ goes to zero:

= 3x^2 + 0 + 0

\boxed{f'(x) = 3x^2}

Why This Works

The limit definition captures the slope of the tangent at a point. We’re asking: as the second point gets infinitely close to the first, what does the slope of the secant line approach?

When we expand $(x+h)^3$ and cancel the $x^3$ , we’re left with terms that are either linear in $h$ or higher powers of $h$ . After dividing by $h$ , only the constant term (with respect to $h$ ) survives the limit — everything else vanishes.

This is exactly why the power rule $\frac{d}{dx}x^n = nx^{n-1}$ works: the binomial expansion always produces one term with $h^1$ coefficient (which survives) and the rest with higher powers of $h$ (which vanish). First principles just makes this explicit.

Alternative Method

Once you understand first principles, use the Power Rule directly in exams where it’s allowed: if $f(x) = x^n$ , then $f'(x) = nx^{n-1}$ .

For $x^3$ : bring down the exponent, reduce it by 1 — $\frac{d}{dx}x^3 = 3x^{3-1} = 3x^2$ .

Takes 5 seconds. But CBSE specifically says “from first principles” when they want the limit method, so read the question carefully.

Common Mistake

Forgetting to divide by h before taking the limit.

Students often write:

f'(x) = \lim_{h \to 0} (3x^2h + 3xh^2 + h^3)

…and then substitute $h = 0$ directly to get $0$ . That’s wrong — you must cancel the $h$ in the denominator first.

The expression $\frac{f(x+h) - f(x)}{h}$ is a $\frac{0}{0}$ form at $h=0$ . The whole point of the algebra is to eliminate that $h$ from the denominator so the limit is no longer indeterminate.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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