Differentiation — implicit, logarithmic, and inverse-trig (IB Math AA HL)

Key ideas

• Implicit: d/dx(y²) = 2y dy/dx and d/dx(xy) = y + x dy/dx (product rule). Collect the dy/dx terms on one side, then factorise.
• Logarithmic: for y = xˣ, write ln y = x ln x, then (1/y) dy/dx = ln x + 1, so dy/dx = xˣ(ln x + 1).
• d/dx(aˣ) = aˣ ln a, and d/dx(logₐ x) = 1/(x ln a).
• d/dx(arcsin x) = 1/√(1 − x²), d/dx(arccos x) = −1/√(1 − x²) and d/dx(arctan x) = 1/(1 + x²). Use the chain rule for arctan(2x) and so on.
• Also new at HL: sec x, cosec x and cot x, with d/dx(tan x) = sec² x and d/dx(sec x) = sec x tan x.
• The gradient of a tangent from implicit differentiation needs both coordinates of the point, so substitute x and y together.

Worked example

The curve C has equation x² + xy + y² = 7. Show that the point (1, 2) lies on C, and find the equation of the tangent to C at that point.

1. Check the point: 1² + 1 × 2 + 2² = 1 + 2 + 4 = 7, so (1, 2) is on C.
2. Differentiate each term with respect to x: 2x + (y + x dy/dx) + 2y dy/dx = 0.
3. Collect the dy/dx terms: (x + 2y) dy/dx = −(2x + y), so dy/dx = −(2x + y)/(x + 2y).
4. At (1, 2): dy/dx = −(2 + 2)/(1 + 4) = −4/5.
5. Tangent: y − 2 = −(4/5)(x − 1), which is 4x + 5y = 14.

Answer: Gradient −4/5; tangent 4x + 5y = 14.

Exam-style HL AA questions on this topic

• Advanced Differentiation questions — 43 questions with mark schemes, plus free worked examples
• Differential Equations and Maclaurin questions — 62 questions with mark schemes, plus free worked examples
• All HL AA Calculus questions by topic

See it on a graph

• Graph of y = x³ − 3x
• Intersections and turning points
• Open the free Grapher

Maths IA ideas that use differentiation

• How fast is a sprinter at each moment? Differentiating 100 m splits
• Designing a smooth roller-coaster drop with piecewise functions
• Why aren't drinks cans the optimal shape?
• What is the best shape for a rain gutter?
• All HL AA IA ideas

Related in Calculus

• Integration by parts and cyclic integrals
• Maclaurin series and approximations
• Differential equations — separable and integrating factor

Revise Differentiation for the exam

• HL AA revision notes — every unit summarised, with the key results to learn
• HL AA past papers — specimen papers and timed practice papers
• Formulas for Calculus — on the one-page HL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original calculus and functions problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S128. All 24 →

Formulas in this topic: Chain rule · Derivative of sin x · Product rule · Quotient rule. All formulas A–Z

Sibling topic in other IB Maths courses