Practice Flashcards
What is the power rule for d/dx(xⁿ)?
Track your progress — Sign up free to save your progress and get smart review reminders based on spaced repetition.
All Flashcards in Topic 5.9
Below are all 24 flashcards for this topic. Sign up free to track your progress and get personalized review schedules.
5.9.18 cards
What is the power rule for d/dx(xⁿ)?
n·x^(n−1) — bring the power down as a multiplier, then reduce the power by 1.
Derivative of a constant?
0 — a constant graph is a flat horizontal line, so its gradient is 0.
d/dx(sin x) and d/dx(cos x)?
sin x → cos x; cos x → −sin x (mind the minus). x must be in radians.
d/dx(eˣ)?
eˣ — the exponential function is its own derivative.
d/dx(ln x)?
1/x.
How do you differentiate √x?
Rewrite as x^(1/2); then it becomes ½x^(−1/2) = 1/(2√x).
How do you find the gradient of f at x = a?
Compute f′(x), then substitute: gradient = f′(a).
Equation of the tangent at x = a?
y − f(a) = f′(a)(x − a), where f′(a) is the gradient and (a, f(a)) is the point.
5.9.28 cards
State the chain rule.
dy/dx = dy/du · du/dx — differentiate the outer function, then multiply by the derivative of the inside.
d/dx of e^{kx}?
k·e^{kx} (chain rule: the k is the derivative of the inside kx).
d/dx of (ax + b)ⁿ?
n·a·(ax + b)^(n−1) — power rule on the outside, times a (the inside derivative).
State the product rule.
(uv)′ = u′v + uv′ — first times derivative of second, plus second times derivative of first.
State the quotient rule.
(u/v)′ = (u′v − uv′)/v² — mind the minus and the order u′v − uv′.
How do you decide which rule to use?
Fraction → quotient; two factors multiplied → product; a function inside a function → chain.
Differentiate y = x·eˣ.
Product rule: (1)eˣ + x·eˣ = eˣ(1 + x).
d/dx of sin(3x)?
3cos(3x) (chain rule: derivative of the inside 3x is 3).
5.9.38 cards
Chain rule for related rates?
dy/dt = (dy/dx)(dx/dt) — linked quantities have linked rates.
Circle: how does area rate link to radius rate?
A = πr² ⇒ dA/dt = 2πr · dr/dt.
Sphere: how does volume rate link to radius rate?
V = (4/3)πr³ ⇒ dV/dt = 4πr² · dr/dt.
The 4-step related-rates recipe?
Write the link → differentiate it → apply the chain rule with the known rate → substitute the value and solve.
When do you substitute the given value (e.g. r = 5)?
LAST — after differentiating. Substituting early removes the variable you need to differentiate.
A balloon's volume grows at 100 cm³/s; find dr/dt at r = 5.
100 = 4π(25)dr/dt ⇒ dr/dt = 1/π ≈ 0.318 cm/s.
What does a NEGATIVE rate mean in a related-rates problem?
The quantity is decreasing (e.g. melting, draining) — keep the minus sign.
If dr/dt = 0 at an instant, what is dA/dt for A = πr²?
dA/dt = 2πr·dr/dt = 0 — the area is momentarily not changing.
Topic 5.9 study notes
Full notes & explanations for Differentiation rules (HL only)
Math AI exam skills
Paper structures, command terms & tips
Want smart review reminders?
Sign up free to track your progress. Our spaced repetition algorithm will tell you exactly which cards to review and when.
Start Free