The differentiation rules, and when to use each

Differentiation in Add Math isn't really one skill, it's a set of tools, each suited to a different shape of expression. Students who already know all four rules individually still lose marks when they apply the wrong one, or reach for the chain rule where the product rule was needed.

Under analytic marking, a correctly chosen rule applied with a small algebra slip usually earns more method marks than the wrong rule applied perfectly. That makes rule identification, not just rule mechanics, the real skill worth building.

The power rule handles any term of the form $x^n$, and it's the rule every other differentiation technique eventually reduces to.

$$\frac{d}{dx}\left(x^{n}\right) = nx^{n-1}$$

This covers polynomial terms directly, and, with a small extension, negative and fractional indices too, so $\frac{1}{x^{2}}$ differentiates the same way once it's rewritten as $x^{-2}$.

• Use it directly on any single power term, or on a sum of power terms, differentiating term by term.
• Rewrite roots and fractions in index form first, $\sqrt{x} = x^{1/2}$, $\frac{1}{x^{3}} = x^{-3}$, before applying the rule, rather than trying to differentiate the original form directly.
• This rule alone is not enough once an expression involves a function inside another function, or two functions multiplied or divided together, that's where the next three rules come in.

The chain rule applies whenever one function is nested inside another, a composite function, like $(3x^{2}+1)^{5}$ or $\sin(2x)$, where the "inside" expression is itself a function of $x$.

$$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$

How to spot when the chain rule is needed

The clearest sign is a bracket, or an inner expression, raised to a power, or sitting inside a trigonometric, exponential, or logarithmic function. If the expression can be described as "something to the power of, or a function of, something else that isn't just $x$", the chain rule is the tool.

• Set $u$ as the inner expression, differentiate the outer function with respect to $u$, then multiply by $\frac{du}{dx}$.
• A very common slip is forgetting the final multiplication by $\frac{du}{dx}$, the derivative of the outer function alone is not the answer.
• The chain rule also underlies rates-of-change questions, where two related quantities change with respect to time, and connecting their rates needs exactly this same structure.

The product rule applies when an expression is two separate functions of $x$ multiplied together, such as $x^{2}\sin x$ or $(2x+1)(x^{2}-3)$, note that this second example could also be expanded first and differentiated term by term, which is often the simpler approach when the brackets are easy to expand.

$$\frac{d}{dx}\left(uv\right) = u\frac{dv}{dx} + v\frac{du}{dx}$$

• Identify the two separate functions being multiplied and label them $u$ and $v$ clearly before differentiating either one, this small step prevents most of the errors students make with this rule.
• Differentiate each function separately first, then substitute into the formula, rather than trying to do both steps at once.
• A common mistake is multiplying the two derivatives together ($\frac{du}{dx} \times \frac{dv}{dx}$) instead of following the correct structure, this is the single most frequent error with the product rule.

The quotient rule applies when an expression is one function of $x$ divided by another, such as $\frac{x^{2}}{2x-1}$ or $\frac{\sin x}{x}$.

$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^{2}}$$

• Label the numerator $u$ and the denominator $v$ before starting, exactly as with the product rule.
• The order in the numerator of the formula matters, $v\frac{du}{dx} - u\frac{dv}{dx}$, not the reverse, since subtraction isn't commutative and reversing the order gives the negative of the correct answer.
• The denominator is always $v^{2}$, the square of the original denominator, not $v$ itself.
• If the denominator is a simple constant or a single power of $x$, it's often quicker to rewrite the expression as a product (multiplying by a negative power) and use the product rule instead.

Under time pressure, it helps to have a quick, consistent way to decide which rule an expression needs, rather than guessing.

1. 1

   Look at the overall shape first

   Is it a single term, a sum of terms, two things multiplied, two things divided, or a function nested inside another? This decides which rule applies before any algebra begins.

2. 2

   Simplify where possible

   Expand brackets that are simple polynomials, or rewrite roots and fractions in index form, since this can turn what looks like a product or quotient rule question into a straightforward power rule one.

3. 3

   Label u and v (or u and the outer function) clearly

   For product, quotient, or chain rule questions, writing out u and v (or u) before differentiating anything reduces careless errors significantly.

4. 4

   Differentiate each labelled piece separately

   Find each derivative on its own, check it, and only then substitute into the relevant formula, trying to do it all in one line is where most mistakes happen.

5. 5

   Substitute carefully into the correct formula

   Double-check the formula's structure, especially the subtraction order in the quotient rule, before finalising the answer.

Differentiation is one of the chapters where seeing a student's actual working matters more than most, because the mistakes tend to be about rule choice and setup rather than arithmetic, the kind of error that's hard to spot from a final answer alone but obvious the moment someone watches the working unfold. A teacher working one-to-one can catch exactly where a student defaults to the wrong rule, or forgets a chain-rule multiplication, and correct the habit before it repeats across dozens of practice questions.

Our teachers are experienced; lessons are taught online, in English, while the SPM papers themselves are set bilingually in Bahasa Melayu and English.

If differentiation rule choice is a recurring source of lost marks, a one-hour paid trial class, at the teacher's own rate, from RM50 per hour depending on experience, is a reasonable way to see whether working through this chapter with a teacher watching your process helps. We won't promise a particular outcome, but this is exactly the kind of pattern that's easier to fix with a second pair of eyes on the actual working.