The power rule differentiates single terms, the product rule handles two factors multiplied together, and the quotient rule handles one expression divided by another. Choosing the right rule is half the work.
This lesson opens SPM Additional Mathematics differentiation. Next comes applying the chain rule.
The power rule
If y = axⁿ, then dy/dx = anxⁿ⁻¹. Multiply by the power, then reduce the power by one.
Example: y = 4x³ − 5x² + 7x − 2 gives dy/dx = 12x² − 10x + 7. A constant term disappears, and x on its own becomes 1.
Roots and fractions must be rewritten first. For y = 3 ÷ x² + √x, write y = 3x⁻² + x^(1/2). Then dy/dx = −6x⁻³ + (1/2)x^(−1/2), which is −6 ÷ x³ + 1 ÷ (2√x).
The product rule
If y = uv, then dy/dx = u′v + uv′.
Example: y = x²(3x + 1). Let u = x² and v = 3x + 1, so u′ = 2x and v′ = 3. Then dy/dx = 2x(3x + 1) + x²(3) = 6x² + 2x + 3x² = 9x² + 2x.
Check by expanding: y = 3x³ + x², so dy/dx = 9x² + 2x. The two routes agree.
The quotient rule
If y = u ÷ v, then dy/dx = (u′v − uv′) ÷ v².
Example: y = (2x + 1) ÷ (x − 3). Let u = 2x + 1, v = x − 3, so u′ = 2 and v′ = 1.
dy/dx = (2(x − 3) − (2x + 1)(1)) ÷ (x − 3)² = (2x − 6 − 2x − 1) ÷ (x − 3)² = −7 ÷ (x − 3)².
Which rule should you choose?
| The expression is | Use |
|---|---|
| A sum of terms such as ax^n | Power rule term by term |
| Two expressions multiplied, long to expand | Product rule |
| One expression divided by another with x in both | Quotient rule |
| A bracket raised to a power | Chain rule |
The mistake that costs marks
The common error is to differentiate each factor and multiply the answers.
| Step | Wrong | Right |
|---|---|---|
| d/dx of x²(3x + 1) | 2x × 3 = 6x | 2x(3x + 1) + x² × 3 = 9x² + 2x |
For quotients, the usual slip is reversing the numerator as uv′ − u′v, which changes the sign of the whole answer.
Check yourself
Differentiate y = (x² + 1) ÷ (x − 1), then find the gradient at x = 3.
Answer
u = x² + 1, v = x − 1, so u′ = 2x and v′ = 1.
dy/dx = (2x(x − 1) − (x² + 1)(1)) ÷ (x − 1)² = (2x² − 2x − x² − 1) ÷ (x − 1)² = (x² − 2x − 1) ÷ (x − 1)².
At x = 3: numerator = 9 − 6 − 1 = 2 and denominator = 2² = 4, so the gradient is 1/2.
What to study next
Continue with applying the chain rule. Use the polynomial differentiation tutor to practise the rules, then try the differentiation practice set.
If you would like a teacher to watch you choose between rules on new expressions, see online one-to-one Additional Mathematics tuition.