Differentiate the outside, then times the inside: Real models nest functions: e0.1t is 'e to the power of (something)', and (2x + 1)⁵ is 'something to the power 5'. To differentiate a composite you use the chain rule:
differentiate the outer function (leaving the inside alone), then multiply by the derivative of the inside. Think 'outer derivative × inner derivative'.
IB-style question — chain rule on a power
A population scaled model is y = (3x + 2)⁴.
Find dy/dx.
Step by step
- State the chain rule and name the inside.
- Differentiate the outer power (leave the inside as u): 4u³.
- Differentiate the inside u = 3x + 2.
- Multiply and put the inside back.
Final answer
dy/dx = 12(3x + 2)³.
The two most common chains: Two patterns appear again and again in AI models:
ekx → k·ekx (exponential growth/decay, k constant), and sin(kx) → k·cos(kx) (waves). The extra factor k is just the derivative of the inside.
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Two factors multiplied, or one over another: When two functions are multiplied (like x·eˣ or t·sin t) use the product rule. When one is divided by another (a fraction with x on the bottom) use the quotient rule.
Product: 'first × derivative of second + second × derivative of first'. Quotient: 'bottom × derivative of top − top × derivative of bottom, all over bottom²'.
IB-style question — product rule
A model is f(x) = x·eˣ.
Find f′(x).
Step by step
- Name the two factors and their derivatives.
- Apply (uv)′ = u′v + uv′.
- Factor out eˣ to tidy.
Final answer
f′(x) = eˣ(1 + x).
IB-style question — quotient rule
A concentration model is g(x) = x / (x + 1).
Find g′(x).
Step by step
- Identify top u and bottom v with their derivatives.
- Apply (u/v)′ = (u′v − uv′)/v².
- Simplify the top: (x + 1) − x = 1.
Final answer
g′(x) = 1/(x + 1)². (Always positive, so the model is increasing.)