Chain, product & quotient
Practice Flashcards
What is the derivative of sin x?
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All Flashcards in Topic 5.6
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5.6.19 cards
What is the derivative of sin x?
cos x.
What is the derivative of cos x?
−sin x (note the minus sign).
What is the derivative of eˣ?
eˣ (unchanged).
What is the derivative of ln x?
1/x.
State the chain rule.
d/dx[f(g(x))] = f'(g(x))·g'(x) — outer derivative times inner derivative.
How do you differentiate (ax + b)ⁿ?
n(ax + b)ⁿ⁻¹ × a (multiply by the inner derivative a).
Derivative of sin(3x)?
3cos(3x).
Derivative of e^(4x)?
4e^(4x).
Derivative of ln(2x + 1)?
2/(2x + 1).
5.6.29 cards
State the product rule.
(uv)' = u'v + uv'.
What is the first step in using the product rule?
Label u and v, then find u' and v'.
In words, what is the product rule?
Differentiate the first times the second, plus the first times the derivative of the second.
Is (uv)' equal to u'v'?
No — that's a common error; it's u'v + uv'.
When does a product also need the chain rule?
When one factor is a composite, like e^(2x) or sin(3x).
Differentiate x·eˣ.
1·eˣ + x·eˣ = eˣ(1 + x).
Differentiate x²·sin x.
2x·sin x + x²·cos x.
Why factorise a product-rule answer?
It is usually neater and reveals common factors (e.g. eˣ or a bracket).
Differentiate x·(x + 4) with the product rule.
1·(x+4) + x·1 = 2x + 4.
5.6.39 cards
State the quotient rule.
(u/v)' = (u'v − uv')/v².
What is the order of terms in the numerator?
u'v first, then minus uv' (derivative of top times bottom, minus top times derivative of bottom).
What is the denominator in the quotient rule?
v² — the bottom function squared.
What is the most common quotient-rule error?
A sign error when subtracting uv' (not distributing the minus).
Differentiate (x + 1)/(x − 2).
(1(x−2) − (x+1)(1))/(x−2)² = −3/(x−2)².
Does the quotient rule add or subtract in the numerator?
Subtract: u'v − uv'.
Derivative of (ln x)/x?
(1 − ln x)/x².
Derivative of 1/(x + 1)?
−1/(x + 1)².
What does a 'show that' quotient question require?
Simplifying your derivative to arrive exactly at the printed expression.
Topic 5.6 study notes
Full notes & explanations for Chain, product & quotient
Math AA SL exam skills
Paper structures, command terms & tips
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