Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Calculus — Topic 5.9
📋 Learning Intentions
- State the standard derivatives of \(\sin x\), \(\cos x\), \(\tan x\), \(e^x\), \(a^x\), \(\ln x\), and \(\log_a x\) from memory and from the formula booklet.
- Find the derivative of composite functions involving trigonometric, exponential, and logarithmic expressions using the chain rule.
- Apply the product rule and quotient rule to functions combining these types.
- Calculate the gradient of a curve at a specific point using GDC numerical differentiation and verify algebraically.
- Determine equations of tangent and normal lines to curves defined by trigonometric, exponential, or logarithmic functions.
- Interpret derivatives of these functions in context — rates of change in oscillating, growth, and decay models.
Real-World Introduction
The temperature of a patient recovering from surgery oscillates over a 24-hour cycle. Engineers modelling electrical current in an AC circuit use sine and cosine functions. In both cases, the rate of change of the quantity — how fast the temperature is rising or the current is changing — requires differentiating trigonometric functions.
Exponential and logarithmic functions appear wherever quantities grow or decay: compound interest, radioactive decay, population dynamics, and the spread of disease. The derivative of \(e^x\) is the only function that equals itself — a remarkable property that makes exponential models mathematically tractable and physically meaningful.
🌍 Why This Matters In pharmacokinetics, the concentration of a drug in the bloodstream follows \(C(t) = C_0 e^{-kt}\). The rate at which the drug clears — \(C'(t) = -kC_0 e^{-kt}\) — determines dosing intervals. Without differentiating exponential functions, safe drug administration cannot be calculated.
Core Concepts & Definitions
Before applying differentiation rules, it is essential to recognise which family a function belongs to. Trigonometric, exponential, and logarithmic functions each have their own standard derivative that must be known precisely. These are listed in the IB formula booklet, but exam questions expect fluent application — not just copying.
📖 Key Definition — Standard Derivatives (Formula Booklet) The following derivatives are given in the IB Mathematics AI formula booklet and must be applied correctly:
- \(\dfrac{d}{dx}(\sin x) = \cos x\)
- \(\dfrac{d}{dx}(\cos x) = -\sin x\)
- \(\dfrac{d}{dx}(\tan x) = \dfrac{1}{\cos^2 x} = \sec^2 x\)
- \(\dfrac{d}{dx}(e^x) = e^x\)
- \(\dfrac{d}{dx}(a^x) = a^x \ln