IB Maths AI HL Ahl 5 9 Hl Further Differentiation Questions

Practise differentiating trigonometric, exponential, logarithmic and rational-power functions with chain, product or quotient rules in related-rate problems.

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Differentiate sin x, cos x, tan x, e^x, ln x and x^n for rational n.
Apply chain, product and quotient rules to composite expressions and solve related-rates problems.
Evaluate and interpret the derivative with correct variables, domains and units.

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Study AHL 5.9 (HL)—Further differentiation conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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This question is about applying ideas from logarithms, calculus and probability to an unfamiliar mathematical theory called information theory.
Claude Shannon developed a mathematical theory called information theory to measure the information gained when random events occur. He defined the information, I, that is gained when an event with probability p occurs as

I=-\ln p

where $0<p \leq 1$ . For example, no information is gained ( I=0 ) when an event is certain to $\operatorname{occur}(p=1)$ .

Show, using calculus, that I is a decreasing function of p.

$\frac{\mathrm{d} I}{\mathrm{~d} p}=-\frac{1}{p}$

Note: Award A1 for $\frac{\mathrm{d} I}{\mathrm{~d} p}$ or equivalent (e.g. $I^{\prime}$ ) and A1 for $-\frac{1}{p}$ . Do not accept $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for the first A1 unless followed by $-\frac{1}{x}$ , which can earn A1 A0.
(for $0<p \leq 1$ ) we therefore have $\frac{\mathrm{d} I}{\mathrm{~d} p}<0$
hence the function is decreasing

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IB Maths AI HL Ahl 5 9 Hl Further Differentiation Questions question 1