Shortcuts for derivatives
First principles works, but it’s slow. These rules make differentiation fast.
Power rule
d/dx xⁿ = nxⁿ⁻¹ (for any real n)
| f(x) | f′(x) |
|---|---|
| x⁵ | 5x⁴ |
| x | 1 |
| 7 | 0 |
| x⁻² = 1/x² | −2x⁻³ = −2/x³ |
| √x = x^(1/2) | ½x^(−1/2) = 1/(2√x) |
Rewrite roots and fractions as powers first.
Constant multiple and sum rules
- Constant multiple rule: d/dx [c·f] = c·f′
- Sum rule: d/dx [f + g] = f′ + g′
Example: d/dx (4x³ − 6x + 9) = 12x² − 6
Product rule
(fg)′ = f′g + fg′
Example: y = (x² + 1)(3x − 4)
- f = x² + 1, f′ = 2x
- g = 3x − 4, g′ = 3
- y′ = 2x(3x − 4) + (x² + 1)(3) = 6x² − 8x + 3x² + 3 = 9x² − 8x + 3
Quotient rule
(f/g)′ = (f′g − fg′) / g²
Memory aid: "low d-high minus high d-low, over low squared."
Example: y = (2x + 1)/(x − 3)
- f′ = 2, g′ = 1
- y′ = [2(x − 3) − (2x + 1)(1)]/(x − 3)² = −7/(x − 3)²
Chain rule
For a composite function f(g(x)):
d/dx f(g(x)) = f′(g(x)) · g′(x)
"Derivative of the outside (keep the inside) × derivative of the inside."
Example: y = (3x² + 5)⁴
- Outside: u⁴ → 4u³
- Inside: 3x² + 5 → 6x
- y′ = 4(3x² + 5)³ · 6x = 24x(3x² + 5)³
Example: y = √(1 − x²) = (1 − x²)^(1/2)
- y′ = ½(1 − x²)^(−1/2) · (−2x) = −x/√(1 − x²)
Combining rules
Example: y = x²(2x + 1)³
- Product rule with the chain rule:
- y′ = 2x(2x + 1)³ + x² · 3(2x + 1)² · 2
- = 2x(2x + 1)²[(2x + 1) + 3x] = 2x(2x + 1)²(5x + 1)
Choosing the rule
| Looks like | Rule |
|---|---|
| a power of x | power rule |
| two functions multiplied | product rule |
| one function divided by another | quotient rule |
| a function inside a function | chain rule |
Worked examples
Example 1: d/dx (5x⁴ − 2x³) = 20x³ − 6x².
Example 2: d/dx 3/x = d/dx 3x⁻¹ = −3/x².
Example 3: d/dx (x + 4)⁶ = 6(x + 4)⁵.
Common mistakes
- Multiplying derivatives for a product: (fg)′ ≠ f′g′.
- Order in the quotient rule (f′g − fg′, not the reverse).
- Forgetting the inner derivative in the chain rule.
Try it yourself
- d/dx (x⁷ − 4x² + 1)
- d/dx (2x)(x³ + 1)
- d/dx x/(x + 1)
- d/dx (5x − 2)³
- d/dx √(4x + 1)
Answers: 1) 7x⁶ − 8x 2) 8x³ + 2 3) 1/(x + 1)² 4) 15(5x − 2)² 5) 2/√(4x + 1)
Quick recap
- Power rule, constant multiple rule and sum rule handle polynomials.
- Product rule and quotient rule handle products and quotients.
- Chain rule: outside derivative × inside derivative.