The Logic of the Shortcut#
While First Principles is the “manual” way to find a gradient, the Power Rule is the high-speed shortcut that you’ll use for the rest of Grade 12.
The Rule#
If $f(x) = ax^n$, then:
$$ f'(x) = n \cdot a \cdot x^{n-1} $$Logic: Multiply by the power, then drop the power by 1.
Quick Examples#
| $f(x)$ | $f'(x)$ | Working |
|---|---|---|
| $x^3$ | $3x^2$ | Bring down the 3, power becomes $3-1=2$ |
| $5x^4$ | $20x^3$ | $4 \times 5 = 20$, power becomes $4-1=3$ |
| $7x$ | $7$ | $1 \times 7 = 7$, power becomes $1-1=0$, and $x^0 = 1$ |
| $9$ | $0$ | A constant has no $x$ — its gradient is always 0 |
| $-2x^{-3}$ | $6x^{-4}$ | $(-3) \times (-2) = 6$, power becomes $-3-1=-4$ |
1. The Sum/Difference Rule#
You can differentiate term by term:
$$ \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) $$Example#
$f(x) = 3x^4 - 2x^3 + 5x - 1$
$f'(x) = 12x^3 - 6x^2 + 5$
Each term is differentiated independently. The constant $-1$ disappears.
2. Preparing for the Power Rule#
You cannot use the power rule until every term is in the form $ax^n$. Most exam questions are “traps” that require rearrangement first.
The 3 Forbidden Forms#
Form 1: $x$ in the Denominator#
Move it up using negative exponents.
| Before | After |
|---|---|
| $\frac{1}{x}$ | $x^{-1}$ |
| $\frac{3}{x^2}$ | $3x^{-2}$ |
| $\frac{5}{2x^3}$ | $\frac{5}{2}x^{-3}$ |
Example: Differentiate $f(x) = \frac{4}{x} - \frac{3}{x^2}$
Rewrite: $f(x) = 4x^{-1} - 3x^{-2}$
$f'(x) = -4x^{-2} + 6x^{-3} = -\frac{4}{x^2} + \frac{6}{x^3}$
Form 2: $x$ inside a Root#
Convert to fractional exponents.
| Before | After |
|---|---|
| $\sqrt{x}$ | $x^{\frac{1}{2}}$ |
| $\sqrt[3]{x}$ | $x^{\frac{1}{3}}$ |
| $3\sqrt{x^5}$ | $3x^{\frac{5}{2}}$ |
Example: Differentiate $f(x) = 4\sqrt{x} + \frac{1}{\sqrt{x}}$
Rewrite: $f(x) = 4x^{\frac{1}{2}} + x^{-\frac{1}{2}}$
$f'(x) = 4 \cdot \frac{1}{2}x^{-\frac{1}{2}} + \left(-\frac{1}{2}\right)x^{-\frac{3}{2}} = 2x^{-\frac{1}{2}} - \frac{1}{2}x^{-\frac{3}{2}}$
$= \frac{2}{\sqrt{x}} - \frac{1}{2\sqrt{x^3}}$
Form 3: Brackets/Products#
Expand before differentiating.
Example: Differentiate $f(x) = (2x - 1)(x + 3)$
Expand first: $f(x) = 2x^2 + 6x - x - 3 = 2x^2 + 5x - 3$
$f'(x) = 4x + 5$
3. The “Split the Fraction” Technique#
When you have a fraction with multiple terms in the numerator, divide each term by the denominator separately.
Example: Differentiate $y = \frac{x^3 - 4x^2 + 2}{x^2}$
Split: $y = \frac{x^3}{x^2} - \frac{4x^2}{x^2} + \frac{2}{x^2} = x - 4 + 2x^{-2}$
$\frac{dy}{dx} = 1 + 0 - 4x^{-3} = 1 - \frac{4}{x^3}$
4. Notation#
Both notations mean the same thing:
| Notation | Meaning |
|---|---|
| $f'(x)$ | The derivative of $f(x)$ |
| $\frac{dy}{dx}$ | The derivative of $y$ with respect to $x$ |
| $D_x[f(x)]$ | The derivative operator applied to $f(x)$ |
Use whichever the question uses. If the function is given as $y = ...$, answer with $\frac{dy}{dx}$. If it’s $f(x) = ...$, answer with $f'(x)$.
5. Full Worked Exam-Style Question#
Determine $f'(x)$ if $f(x) = \frac{(x+2)^2}{\sqrt{x}}$
Step 1: Expand the numerator: $(x+2)^2 = x^2 + 4x + 4$
Step 2: Rewrite the denominator: $\sqrt{x} = x^{\frac{1}{2}}$
Step 3: Split the fraction: $f(x) = \frac{x^2}{x^{\frac{1}{2}}} + \frac{4x}{x^{\frac{1}{2}}} + \frac{4}{x^{\frac{1}{2}}}$
$= x^{\frac{3}{2}} + 4x^{\frac{1}{2}} + 4x^{-\frac{1}{2}}$
Step 4: Differentiate: $f'(x) = \frac{3}{2}x^{\frac{1}{2}} + 4 \cdot \frac{1}{2}x^{-\frac{1}{2}} + 4 \cdot \left(-\frac{1}{2}\right)x^{-\frac{3}{2}}$
$= \frac{3}{2}\sqrt{x} + \frac{2}{\sqrt{x}} - \frac{2}{\sqrt{x^3}}$
🚨 Common Mistakes#
- Differentiating too early: You must change EVERY term into $ax^n$ form before applying any rules. Don’t differentiate half the expression while the other half still has roots or denominators.
- Fractional exponent arithmetic: $\frac{1}{2} - 1 = -\frac{1}{2}$, NOT $-\frac{3}{2}$. And $-\frac{1}{2} - 1 = -\frac{3}{2}$. Be precise with fraction subtraction.
- Forgetting to expand brackets: You cannot differentiate $(x+2)(x-3)$ directly. You MUST expand first.
- Wrong notation: If the question gives $y$, use $\frac{dy}{dx}$. If it gives $f(x)$, use $f'(x)$. Mixing these up loses marks.
- Leaving the answer in negative exponents: Unless the question asks for it, convert $x^{-2}$ back to $\frac{1}{x^2}$ for a cleaner final answer.
💡 Pro Tip: The “Species” Rule#
You can only differentiate terms separated by PLUS or MINUS. If you have $\frac{x^2 + x}{x}$, you must divide each term by $x$ first to “split the species” ($x + 1$) before you differentiate.
⏮️ First Principles | 🏠 Back to Calculus | ⏭️ Tangent to a Curve