Short answer: look at the last operation you would do if you calculated the function by hand. If it is multiplication, use the product rule. If it is division, use the quotient rule. If you apply a function to something other than plain x (a power, sin, e…, ln), use the chain rule. Many exam questions need more than one rule.
The three rules
- Product: (f·g)′ = f′g + fg′
- Quotient: (f/g)′ = (f′g − fg′) / g²
- Chain: [f(g(x))]′ = f′(g(x)) · g′(x), meaning "derivative of the outside, times derivative of the inside"
The "last operation" test
Imagine plugging in x = 2 and working out the value. The final step tells you the outer rule:
- x²·sin x: you multiply last → product rule.
- (3x + 1)5: you raise to the 5th power last → chain rule.
- ex / (x² + 1): you divide last → quotient rule.
- sin(x²): you take sine last → chain rule.
Worked example: rules combined
Differentiate y = x²·sin(3x).
The last operation is multiplication, so start with the product rule. f = x², f′ = 2x. g = sin(3x), and g′ needs the chain rule: cos(3x)·3.
y′ = 2x·sin(3x) + 3x²·cos(3x).
Shortcuts that save time
- Constant denominators: (x³ + x)/5 needs no quotient rule. Just multiply by 1/5.
- Rewrite as a power: 1/x² = x−2, and its derivative is −2x−3. That is faster than the quotient rule.
- Simplify first: (x² + x)/x = x + 1 for x ≠ 0.
Mistakes that cost marks
- Forgetting the inside derivative: (sin 3x)′ is 3 cos 3x, not cos 3x.
- Getting the quotient rule order backwards. The numerator is f′g − fg′, and the order matters because of the minus sign.
- Writing (f·g)′ = f′·g′. That is never the product rule.
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